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You are given the system of two equations:
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5x+4y=-1
25x+20y=-5
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and are asked how many solutions this system has. Just the way this question is asked might suggest that there could be something unusual involved.
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You can tell the number of solutions by understanding how many points in common the graphs of the two equations have in common. There are three possibilities as follows:
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(1) The linear (straight line) graphs for each equation have different slopes and therefore, these graphs cross at only one point. The coordinate pair for that point is the only solution to such a system. or
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(2) The linear (straight line) graphs for each equation have the same slope and this presents two possibilities. First, either the graphs are separate parallel lines (like railroad tracks) and they never cross. Therefore there are no common solutions. And second, it could be possible that the two graphs lie on top of each other, so that every possible solution for one equation is also a solution for the other. In such a case there are an infinite number of common solutions for the two equations.
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This problem is an example of the last possibility. The two equations have an infinite number of common solutions. How can you tell? Look again at the two equations:
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5x+4y=-1
25x+20y=-5
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If you multiply the top equation (all terms on both sides) by 5 you do not change the equation. However, when you do the multiplication by 5 the top equation becomes:
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25x + 20y = -5
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Notice that this is identical to the bottom equation. Therefore, the two equations have the same graphs. That means that the graph of the solution points for the top equation lies on top of the graph of the solution points for the bottom equation. This tells you that the graphs have an infinite number of common solution points which translates to an infinite number of common solutions.
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I hope that this helps you to understand the three possible solutions that a system of two linear equations can have. And how you can picture the graphs of each equation to help you understand the number of points that will be solutions for a particular system of linear equations.
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Let N represent the unknown number.
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By definition the reciprocal of the number is
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So the sum of the unknown number and its reciprocal is:
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The problem tells you that this sum is equal to 5/4 of the number meaning that it is equal to 5/4 times the number. This can be written as
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So we can write the equation that says this as:
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We can now get rid of the N and the 4 in the denominator bys multiplying both sides of this equation (all terms) by 4*N as follows:
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Cross out the denominators with the corresponding terms in the numerator as follows:
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With these cancellations we are left with:
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And doing the multiplications in each of the terms simplifies this equation to:
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Collect the terms containing N on one side of this equation by subtracting from both sides of the equation. This subtraction eliminates the on the left side and the resulting equation is:
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Doing the subtraction on the right side simplifies this to:
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Now solve for N by taking the square root of both sides:
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After taking the square root of both sides you are left with two answers. Either:
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or
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The +2 and the -2 are possible values for N because in either case, if you square them to find N-squared you get +4 as the answer. So we now have that N, the unknown number, is either +2 or -2.
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Check these two answers by one at a time adding the number to its reciprocal and seeing if that results in 5/4 times the number as follows:
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or:
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And if you work these two equations out you will find that in each of them, the right side is equal to the left side. This means that our two answers are correct. N can be either +2 or -2.
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Hope this helps you to understand the problem a little better.
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There's not enough information here to tell where the difficulty was. It could have been a math error. But maybe more important is that you need to remember that in using your method you actually found the volume of the sand plus the volume of the space between the grains of sand.
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Maybe the exercise required that you are only interested in the actual volume of all the grains of sand, meaning that you must not count the space between the irregular shaped grains of sand. If that is the case, then you might want to use the Archimedes principle.
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In times long gone by, Archimedes discovered a means of determining the volume of irregular shaped bodies. It is based on the principle that water and an irregular shaped item cannot both occupy the same space at the same time. When the irregular shaped item is put into water, it will displace a volume of water that is equal to its volume. So here is a way that you might have been expected to solve this problem.
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First partially fill the cylindrical bin with just water, but it must be enough to cover the sand that will be added later. Mark the level of the water in the bin. Then pour the sand into the water and this will cause the level of the water to rise as the sand displaces it. When all the sand has been poured into the bin, mark the new level of the water. The volume of the water between the two marks (its original level and its level after all the sand has been poured into it) is equal to the actual volume of just the grains of sand.
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You could then calculate the volume of the water displaced by finding the circular area of the bin (pi times the square of the radius of the cylinder) and multiplying that by the distance between the two marks. (Be sure to have the radius and the distance between the two marks in the same unit of measure.) And this volume of water would equal the volume of just the grains of sand.
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Or you could pour water from the circular bin (being careful not to pour any sand out) into a container that is calibrated for measuring the volume of liquids. Continue pouring until the water has returned to the level of the first mark you made prior to pouring the sand into it. The calibrated container will then tell you the volume of the displaced water and, again, that volume will be equal to the volume of just the grains of sand.
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Maybe this will help. If it's not what you are looking for, post your question again and maybe another tutor will be able to help you.
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Let X and Y represent the two unknown numbers. Since we have two unknowns, we need two independent equations to solve for them.
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The problem tells you that the sum of these two numbers equals 77. So let's write an equation that says that:
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X + Y = 77
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The problem also tells you that the difference between the two numbers is 7. Now let's write an equation that says that:
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X - Y = 7
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These are the two equations we need, and they will allow us to find values for X and Y. One of the ways we can solve a pair of equations such as this is by variable elimination. If we can add or subtract the two equations and in doing so one of the variables disappears, we can solve the new equations for the other variable. Let's write the equations one above the other like this:
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X + Y = 77
X - Y = 7
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Notice that if we add the two equations vertically that the +Y and the -Y cancel each other so they disappear. The X in the top equation adds to the X in the lower equation to give 2X and on the other sides of the two equations the 77 adds to the 7 to give 84. So after adding the two equations vertically we are left with a new equation:
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2X = 84
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We now solve for X by dividing both sides of this new equation by 2 to get:
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X = 42
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Next we can go back to either of the two original equations we wrote and in the one that we choose, we can substitute 42 for X and then solve for Y. Let's select the first of our two original equations:
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X + Y = 77
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Substitute 42 for X and this becomes:
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42 + Y = 77
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Solve for Y by subtracting 42 from both sides of this equation. When we do that the 42 on the left side disappears and we are left with:
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Y = 77 - 42 = 35
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That means that we know the two unknown numbers are 42 and 35. You can check to verify this by adding them to ensure that their total is 77 and then subtracting them to see that the difference is 7.
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Hope this helps you to understand the problem a little more.

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The total of her four scores is the sum of the three known scores (25 + 15 + 12) plus x, her unknown score on the fourth assessment task. If you divide that total by 4 (which is the number of scores) you will get the average score on the four assessment tasks. The problem tells you that the average score is 14.5
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So let's set this up as an equation. Begin by writing the sum of the four scores:
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Divide that sum by 4 to find the average:
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And since we know that the average is 14.5 we can set this equal to 14.5:
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Now we can solve this for x. Begin by adding the first three scores of 25, 15, and 12 to get a total of 52. This makes the equation become:
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Then we can get rid of the denominator 4 on the left side by multiplying both sides of the equation by 4 as follows:
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On the left side, the 4 in the numerator cancels with the 4 in the denominator, and on the right side the product of 4 times 14.5 is 58. This reduces the equation to:
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Next you can get rid of the 52 on the left side by subtracting 52 from both sides to get:
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That's the answer to this problem. She scored 6 on the fourth assessment task.
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So, if she scored a 6 on the fourth assessment task, the resulting average should be 14.5.
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Let's check that. By scoring an 6 on the final assessment task, the four scores become 25 + 15 + 12 + 6. Add these 4 scores and the total is 58. Then find the mean by dividing the total 58 points by 4 and the answer is 14.5, so the answer checks.
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Hope this helps you to understand the problem (and also see that she didn't prepare enough so that she did well on the fourth assessment task).
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You have two unknowns, the number of daytime hours (call them D) and the number of nighttime hours (call them N). This means that you will need two independent equations to solve for these two unknowns. So let's go through the problem and see if we can find two such equations.
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First, we know that D plus N is the total number of hours that were spent using the computer. The problem tells you that this total was 28 hours. Therefore, we can write one of the equations as follows:
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D + N = 28
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Next the problem tells you that the bill for using the computer was $411. But this bill came from using the computer both during the day and also during the night. Each hour during the day that the computer was used cost $30. So the total cost for daytime use can be found by multiplying $30 times D, the number of hours it spent on daytime use. This can be written as 30D. Similarly, the total spent on using the computer at night was $10.50 per hour times N, the number of hour of nighttime use. This can be written as 10.5N. The addition of these two costs (daytime plus nighttime) equals the total $411 cost of using the computer. So we can write the second equation as:
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30D + 10.5N = 411
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Now all that needs to be done is to solve this set of two equations to find the values of D and N. One way to do this is by using substitution. Look at the first equation, and let's solve it for one of the variables in terms of the other. For example, let's subtract D from both sides of this equation. This will get rid of the D on the left side of the equation and a minus D will appear on the right side. Therefore, this equation becomes:
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N = 28 - D
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Now we know that N equals 28 - D, we can go to the second equation and replace N in it with its equivalent 28 - D. When we do that, the second equation becomes:
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30D + 10.5*(28 - D) = 411
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Do the distributed multiplication on the left side by multiplying 10.5 times each of the two terms in the parentheses. This results in the equation becoming:
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30D + 294 - 10.5D = 411
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On the left side combine the two terms that contain D by subtracting 10.5D from 30D to get:
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19.5D + 294 = 411
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Next, get rid of the 294 on the left side by subtracting 294 from both sides. The resulting equation is:
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19.5D = 117
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Finally, solve for D by dividing both sides by 19.5 to get:
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D = 117/19.5 = 6
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So we now know that 6 hours was spent using the computer during daytime hours. And since the first equation tells us that a total of 28 hours was spent using the computer, the remaining 22 hours must have been for nighttime use. Let's check that by finding what the cost of such usage would be.
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For 6 hours of daytime use at $30 per hour the cost would be 6*$30 = $180. And for 22 hours of nighttime use at $10.50 per hour the cost would be 22*$10.50 = $231. So the total cost would be $180 + $231 and this equals $411 which is what the problem says it should be.
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The answer therefore is that the computer was used 6 hours during the day and 22 hours at night.
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Hope that this discussion helps you to understand better how to determine the number of equations you need for problems such as this one (the number of equations needed equals the number of unknown variables), and once you have the equations, one way that you can solve for the unknown variables.
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You can work this problem by starting at the back and moving toward the front. The term "a number" tells you that there is a number of unknown value. You can use X to represent that number.
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Then there is the wording "8 times a number" which you can translate to 8 times X or just 8X.
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Then moving closer to the front you are told "12 more than 8 times a number" which is 12 more than 8X and in algebraic form this is 12 + 8X
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Finally you are told that 28 is less than that. So you can write the inequality:
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28 < 12 + 8X
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You can solve this using rules much like apply to an equation. (An exception to those rules is that if you multiply or divide both sides by a negative number, you reverse the direction of the inequality sign. In this problem, you do not have to worry about that happening.)
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Next, isolate the 8X on the right side by getting rid of the 12 on the right side. Do that by subtracting 12 from both sides to get:
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16 < 8X
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Finally solve the inequality for X by dividing both sides by +8. The result is:
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2 < X
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Since the inequality sign points to the smaller quantity, you can read this as "X is greater than 2" or X > 2. This means that on the number line X can be any value to the right of +2.
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Hope this helps you to understand this inequality problem.
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Your answer is correct. You basically worked it as a proportion. The critical feature you recognized is that if six-thirteenths could discriminate, then the remaining seven-thirteenths could not. And the problem tells you that 28 could not. So you can establish the following proportion:
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Seven-thirteenths is to 28 as six-thirteenths is to X
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In which X represents the number of audiophiles that could discriminate. In proportion form this can be written as:
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This proportion can be solved by cross-multiplying. Multiply the numerator on the left side by the denominator on the right side. Set this equal to the product of the numerator on the right side times the denominator on the left side. In equation form this is:
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You can solve this for X by dividing both sides by which is the multiplier of the X. Remember that if you divide by a fraction, it is the same as inverting the fraction and multiplying. In other words we can multiply both sides of the equation by . This makes the equation become:
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On both sides you have a 13 in the numerator that will cancel with a 13 in the denominator as follows:
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and you are left with:
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Obviously on the left side you have just X and on the right side you can do exactly what you did ... namely divide 7 into 28 and multiply that result times 6 to get your answer of 24.
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This may help you to see how to do it as an equation, but you did the analysis and math manipulations exactly correct. Good job!!! You knew what you were doing and how to solve this problem. Keep up the good work.
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Here's a way to think about this problem. In an equation such as this one you are trying to graph all the coordinate pairs of points [that is (x, y) points) that are solutions to the problem. In an equation such as this one, you can choose a value for x and substitute that value into the equation and solve for the value of y that corresponds to that value of x.
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For example, you could choose x = 3. Substitute that value of x into the equation as follows:
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7*3 + 14y = 10
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Multiplying the 7 times 3 makes the equation become:
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21 + 14y = 10
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Subtract 21 from both sides and you have:
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14y = -11
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Solve for y by dividing both sides by 14, and you get:
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y = -11/14 = -0.7857 ... rounded to 4 decimal places
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This tells you that the coordinate point (3, -0.7857) is on the graph and is a solution to the equation.
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This is just an example to show you what is going on when you choose values for one of the unknowns in an equation.
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Now think of the process of graphing. Picture in your mind the origin and the horizontal x-axis and the vertical y-axis. Now picture a point on the y-axis. What has to be the corresponding value of x? Any point on the y-axis must have a corresponding value of x equal to zero. So the point where the graph crosses the y-axis (the y-intercept) can only be occur where x equals zero.
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Just for further clarification, think of the plotting the (x, y) point (1, 2) on a coordinate system. You move from the origin to the right one unit on the x-axis and then go up 2 units above it in the y direction. This point is not on the y-axis and therefore is not the y-intercept. Similarly plot the point (-5, 8). Starting at the origin, you go 5 units to the left along the x-axis and at that point you go up 8 units in the y direction to the point (-5, 8). This is not on the y-axis either, so it cannot be the y-intercept. Now think about the plotting the (x, y) point (0, -2). You start at the origin and move neither to the left or right along the x-axis because the value of x is zero. Instead you just move down the y-axis 2 units to -2 and mark that point. Note that this point is on the y-axis and is therefore the y-intercept. This illustrates the point that I was trying to make ... whenever the value of x is zero, the corresponding value of y will be on the y-axis and therefore is the y intercept.
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So go to the equation you were given:
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7x + 14y = 10
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Now choose zero for the value of x. When you set x equal to zero, 7 times zero becomes zero and the equation reduces to:
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14y = 10
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And you solve for y by dividing both sides by 14 to get:
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y = 10/14 = 5/7 = 0.7143 ... rounded to four decimal places
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This tells you that the point (0, 0.7143) is on the graph. Because x = 0 the point is on the y-axis, and therefore, 0.7143 is the y-intercept for the graph of the given equation.
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You can just shorten the rule for doing this to say "To find the y-intercept, set x equal to zero and solve for the corresponding value of y. That value for y will be the y-intercept ... the value where the graph crosses the y-axis."
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You can use a similar line of reasoning for finding the x-intercept. Any point on the x-axis will have a corresponding y-value of zero. If y is any value other than zero, the point will be above or below the x-axis and therefore cannot be the x-intercept. The rule becomes "To find the x-intercept, set y equal to zero and solve for the corresponding value of x. That value for x will be the x-intercept ... the value where the graph crosses the x-axis."
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So go to your equation and set y equal to zero as follows:
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7x + 14*0 = 10
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The multiplication of 14 times zero becomes zero and you are left with:
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7x = 10
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Solve for x by dividing both sides by 7 to get:
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x = 10/7 = 1&3/7 = 1.4286 ... rounded to 4 decimal places
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So we have the (x, y) point of (1.4286, 0). This point satisfies the equation and is on the x-axis because the corresponding value of y is zero. Therefore, 1.4286 is the x-intercept.
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In summary, the y-intercept is 0.7143 and the x-intercept is 1.4286
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Now that you know the two intercepts, you can mark them on the coordinate system and connect them with a line through them. Then any point on that line is an (x, y) point in which the value of x and its corresponding value of y will satisfy the equation. You should get a graph that looks like this:
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Recall way back at the beginning of this problem we worked an example using this equation, letting x = 3, and solving for y to get that the corresponding value of y is -0.7857. You can look at the graph and tell that the point (3, -0.7857 is on the graph.
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I hope that this helps you to understand what you are doing when you set x equal to zero to find the y-intercept and when you set y equal to zero to find the x-intercept. Once you think about it and understand the concept, with a little thought and practice it should become easier. Good luck.
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The first clue in this problem involves the words "a number m." This tells you that you have an unknown number represented by the letter m. And the requirement to "write an equation and solve" means that you will need an equation that involves the letter m in some way, and you will use that equation to solve for the unknown number m. In other words you will work with the equation and in doing so will find the value for m.
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The words "twice a number m" means 2 times m or in shortened form 2m. "four more than twice a number m" therefore just tells you to add 4 to 2m. In algebraic form this is written as 4 + 2m. And the words "is" means equals. Finally, "the sum of 4 and 8" is 4 + 8.
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Putting this all together we can write the equation:
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4 + 2m = 4 + 8
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You now want the term 2m (that is the term that contains the unknown) to be by itself on the left side of the equation and all the numbers to be grouped on the right side. So you need to get rid of the 4 on the left side. To do that, just subtract 4 from the left side. But whatever you do to the left side, you must also do to the right side in order to keep both sides "in balance." In other words subtract 4 from the left side and also subtract 4 from the right side as follows:
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4 - 4 + 2m = 4 - 4 + 8
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When you do the subtracting on both sides, you notice that the +4 and the -4 on each side cancel each other out and you are left with:
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2m = 8
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At this point you can solve for m by dividing both sides by 2 to get:
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m = 8/2 = 4
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So you have written the equation as 4 + 2m = 4 + 8, and you have solved this equation to find that m equals 4.
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Hope this helps you to understand this word problem. Don't let the words confuse you. You just need to break the words in the problem into small groups of words and translate these parts into math terms. Takes some practice, but eventually you'll begin to make sense of word problems. Good luck with your studies.

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Given to expand:
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Here's a long explanation of this problem that might give you some insight into how you do it.
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We are going to define a new term i in which i is the square root of -1. We write it as . That being the case, then if we square both sides of this definition, we also have that . With these two definitions in mind we can proceed with this problem step-by-step.
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Let's work on how we can replace the . Recognize that we can replace -5 by 5 times -1. But by our definitions of i we know that So, substitute for -1 and the 5 times -1 becomes .
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This means that in the first set of parentheses, the term:
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can be replaced by
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Furthermore, under the algebraic rules for square roots, this replacement can be split into two parts as follows:
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But the square root of a squared term, that term being i squared, is just the term itself. So we can replace with just i. With this substitution we then get:
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So going back to the original problem, the contents of the first set of parentheses can be written as:
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It would be a long process to go through this sort of work every time you encounter the square root of a negative number. Instead, you can just use the shortcut of saying "the square root of this negative number, is just the square root of the positive number times i." As an example: for just say it equals times i. [Note that since the square root of 16 is 4, this can be further reduced to ]
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With this short cut in mind, let's look at the contents of the second set of parentheses in the original problem, namely . According to the shortcut, the second term will become just and since the square root of 1 is just 1, the second term is just 1*i or simply i. (This also matches the definition that ). So the second set of parentheses in the problem contains .
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We have now converted the original problem to:
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We can now do the multiplication using the FOIL method. To do that, first multiply the 3 in the first set of parentheses times both of the terms in the second set of parentheses to get the product . Next multiply the second term in the first set of parentheses times both terms in the second set of parentheses. In other words, multiply times to get . But recall from our definitions that . When we replace the by -1, the contents of this product becomes:
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And the multiplication by -1 makes it become:
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Finally, add these two products as follows:
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The the last step is to put this into conventional form by combining the real parts and combining the imaginary parts (all terms containing i) second. This makes the answer become:
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Factor out i from the last two terms and the result is:
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and that's the answer that you said we should get.
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That's the way to do this problem. I hope this gives you some insight into working with complex numbers (numbers having both real and imaginary parts.) It's not too hard, but it takes lots of practice and you need to remember that and that .

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Given to simplify:
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You got the answer correct if you meant it to be interpreted as:
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You correctly recognized that you divide the two in the denominator into the 3 in the numerator. Then, if you recognized that in this problem the x term in the denominator is the divisor and the x term in the numerator is the dividend and when you divide these two you subtract the exponent 3 of the divisor from the exponent -5 of the dividend to get the answer .
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So the answer to the problem could have been written as:
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which is equivalent to
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And by the rules of exponents, you can put the x term in the denominator, but in doing so its exponent changes sign. In this case the answer can be written as the answer that you got. Namely as:
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Hope this helps you to gain some further understanding of and confidence in the method that you used in working this problem. Good job!!! Keep it up!!!
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Given the system of equations to solve:
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2x - y = 6 and
x + 5y = -19
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At this stage of your studies in math, there are a couple of ways that you can solve this system. You can use substitution or you can use variable elimination. Whatever you do, don't try to do too much in your head, because if you make a mistake you won't be able to find what you did wrong.
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So let's try substitution first. Pick one of the equations and solve it for one variable in terms of the other. In this case, it might be easier to select the second equation because we can easily solve it for x in terms of y. In the second equation we can can get rid of the 5y term by subtracting 5y from both sides. This makes the 5y on the left side go away and the right side becomes as shown below:
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x = -19 -5y
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Now with that result from the second equation, we can go to the first equation and substitute -19 - 5y for x. When we replace x in the first equation with (-19 - 5y) the first equation becomes:
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2(-19 - 5y) - y = 6
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Do the distributed multiplication by multiplying 2 times each of the terms in the parentheses and you have:
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-38 - 10y - y = 6
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Get rid of the -38 on the left side by adding +38 to both sides. On the left side adding +38 to -38 results in zero, so the -38 disappears. On the right side adding +38 results in the sum of 38 and 6 which is +44. Therefore, the equation is:
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-10y - y = 44
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Combining the two terms on the left side results in:
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-11y = 44
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And by dividing both sides by -11 you arrive at:
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y = -4
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Finally you can solve for x by going to either of the two equations and substituting -4 for y. Let's go to the second equation and replace y by -4 to make it become:
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x + 5(-4) = -19
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Multiplying the 5 times -4 results in -20 and the equation is then:
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x - 20 = -19
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Get rid of the -20 on the left side by adding +20 to both sides and you get:
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x = -19 + 20 = +1
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The two answers for the unknowns in this system of equations are x = +1 and y = -4.
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Let's check this solution by working the problem in a different way to see if we get the same answer. The second way will involve variable elimination. Start with the two equations:
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2x - y = 6 and
x + 5y = -19
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We're going to subtract (or add) in vertical columns and eliminate one of the variables in doing so. Let's think about eliminating y. We can make y in the first equation equal the y term in the second equation. All we have to do is multiply the first equation (all terms on both sides) by 5. When we do that the first equation becomes:
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10x - 5y = 30
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Let's put the second equation right below it so this pair of equations is:
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10x - 5y = 30 and
x + 5y = -19
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Adding vertically we get: the 10 + x = 11x, the -5y and the +5y cancel each other out, and the 30 and -19 add to give +11. So after adding we are left with 11x on the left side and 11 on the right side as follows:
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11x = 11
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Solve for x by dividing both sides by 11 to get:
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x = 1
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Then you can return to either of the original equations and substitute +1 for x to solve for y. Let's go to the second equation and substitute +1 for x. When we do that the second equation becomes:
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1 + 5y = -19
.
Get rid of the 1 by subtracting 1 from both sides to get:
.
5y = -20
.
and solve for y by dividing both sides by 5. The answer becomes:
.
y = -4
.
So using variable elimination results in x = 1 and y = -4. These are the same results as we got working the problem using substitution, so our answer checks.
.
Hope this helps you to see how you can use these two methods to solve a system of linear equations. All it takes is a little understanding and lots of practice in using these methods.
.

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Why use a chart? If you use a chart, it becomes a recipe for solving such problems without understanding what you are doing. Instead let's analyze the information that you are given and see if we can't think our way through this problem.
.
By reading the problem we know that we have to find out how much volume of a 20% solution we are going to mix with an unknown volume of 70% solution. So we have to find two unknown volumes. Let's call the unknown volume of the 20% solution T and the unknown volume of the 70% solution S. (T stands for 20% and S stands for 70%).
.
The problem says that when we add these two volumes together we end up with 50 liters of a mixed solution. So let's set up an equation for that. When we mix the two unknown volumes we are adding them together, and we want to have the answer be 50 liters. In equation form this is:
.
T + S = 50
.
Since we have two unknowns (T and S) we know that we need two equations to solve for these two unknowns. We already have one of the equations. Now we need to look for a second one.
.
Let's think about how much acid we have in each of the volumes. In T (the volume of the 20% solution) 20% (or 0.2) of it is acid. So if we multiply 0.2 times T that answer should equal the volume of acid in that solution. Similarly, in S (the volume of the 70% solution) 70% (or 0.7) of it is acid. So if we multiply 0.7 times T that answer should equal the volume of acid in that solution. How much of the mixture should be acid? We are told that we are to end up with 50 liters of the mixture, and 40% (or 0.4) of that mixture should be acid. For this we can multiply 0.4 times 50 and get an answer of 20. That volume of 20 liters is to be the volume of the acid in the mixture.
.
So now we can write our second equation that we need. We can add the volume of the acid in the two original solutions and set it equal to the volume of acid in the mixture. In equation form this becomes:
.
0.2T + 0.7S = 20
.
We have now reduced the problem to an algebraic exercise, namely solving for two equations that each have two unknowns. Our two equations are:
.
T + S = 50 and
0.2T + 0.7S = 20
.
You probably already know that we can solve two equations with two unknowns by several methods. For example, we can use variable elimination or substitution. (You will learn some other methods later.)
.
Let's use substitution. From the first equation, we can subtract S from both sides and get:
.
T = 50 - S
.
Then we can go to the second equation and substitute 50 - S in place of T. When we do that substitution, the second equation becomes:
.
0.2(50 - S) + 0.7S = 20
.
We then do the distributed multiplication of the first term by multiplying 0.2 times each of the quantities in the parentheses to get:
.
10 - 0.2S + 0.7S = 20
.
Then we subtract 10 from both sides of the equation to get rid of the 10 on the left side, and the equation we are then left with is:
.
-0.2S + 0.7S = 10
.
By combining the two terms on the left side we have:
.
0.5S = 10
.
and finally we divide both sides by 0.5 and the equation becomes:
.
S = 20
.
This tells us that we need 20 liters of S (the 70% solution) in the mixture. And since the mixture is to be 50 liters, we know that the other 30 liters must come from the 20% solution (T).
.
That's the answer --- 30 liters of 20% solution get mixed with 20 liters of the 70% solution to produce 50 liters of a 40% solution.
.
Hope this helps you to understand how you can think your way through problems such as this one. Just remember that in mixture problems involving two unknowns you need to have two independent equations. (Mixing coffee or nuts of two different prices, or acids of different strengths are common examples).

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It appears as if no tutor has commented to this problem yet. So before it drops
down the line so far that it disappears from the list, I'll offer you some input
that might be useful to you. However, before that, I'm really impressed with the visual
that you provided with your problem. That was quite spectacular and made your statement
of the problem considerably easier to understand. Good job!!!
.
Now to your problem. My first comment is, "Are you sure that the equation:
.

.
describes the parabolic shape of the building?" The reason I ask this comes from
the following analysis:
.
Suppose y equals zero. That should cause the equation to represent the two places
where the graph intersects the horizontal axis on which the bottom of the building
rests. But if you set y equal to zero, the equation becomes:
.

.
Solve this by factoring the right side of the equation as below:
.

.
Notice that the right side of the equation will equal zero also if either of the factors
are zero. So either:
.
or
.
To solve for the value of x that will make the second factor equal to zero, let's first
multiply both sides (all terms) by 36 to get rid of the denominator. This multiplication
results in:
.
which becomes:
.
When you solve this you get the second answer for the intercept as being
.
So the equation you gave for the parabolic arch should intercept the horizontal axis at 0,
as it does, and also at +108, but it doesn't. Instead of 108 it appears to show the right
hand intercept at 150. Is the equation in error or the drawing? Or perhaps I'm do not
understand the problem as well as I should before I comment to it.
.
Regardless of that, here's an idea of how you might determine the lengths that you need.
Look at the drawing you provided of just the single first layer. You need to get the
length of the top horizontal line. Note that it intersects with the parabolic graph on
both ends. Therefore, the endpoints of the line must satisfy the equation for the parabola.
In other words, the endpoints can be solved for x if the value of y is known in the equation
for the parabola. For this first level, what is the value of y for the top horizontal
line? It is 2.5, is it not? So for the given equation you could write the equation as:
.

.
This equation can be solved by first multiplying both sides (all terms) by 36 to get rid
of the denominator. As a result of this multiplication the equation becomes:
.

.
Add x^2 - 108x to both sides and the equation then becomes:
.

.
This is in the standard quadratic form of:
.

.
and therefore it can be solved for the two values of x at the endpoints by applying the
quadratic formula:
.

.
In which you substitute 1 for a, -108 for b, and 90 for c to get:
.

.
This simplifies to:
.

.
and the terms in the radical combine to give:
.

.
Taking the square root results in the two answers of:
.
and
.
and this leads to the two answers for x of 0.839865 and 107.160135.
.
(This is correct for the given equation results in the ground level intercepts are
at 0 and 108, not 0 and 150.)
.
For each following set of x values, all you have to do is increase the value of y by 2.5.
So for the next level up, you would solve the equation:
.

.
This would lead to the quadratic form:
.

.
And in general, you could say that the quadratic solution is:
.

.
which by multiplying out the third term becomes:
.

.
where n is 1, 2, 3, 4, ... up to the ceiling level for the top floor.
.
by applying the quadratic formula you get:
.

.
and by multiplying out the terms you get:
.

.
And then you can divide by 2 to get:
.

.
Now to get the values of x for the multiple levels of the building, just substitute
for n the values of 1, 2, 3, 4, 5, and so on and calculate the pairs of x at each ceiling
level for the entire building. That should enable you to find the triangles of unused space
at each level.
.
I hope this helps you with your problem. Check my math and logic to see if I made some
dumb mistake in doing this analysis rather quickly. If my interpretation of the problem
doesn't help you, please post your problem again and some other tutor may be able to
solve it in a better way than I did.
.

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The basic equation that is used is:
.

.
which we'll shorten to:
.

.
This is pretty straightforward when you think about it. You hop on a bike and ride at 10 mph for 3 hours. How far do you go? 10 mph * 3 hrs = 30 miles of distance.
.
In this train problem we have two separate distances to find because the rates are different. So let's call the train going north Train 1 and the train going south Train 2. (We'll use subscripts to differentiate them.)
.
So, we can write that the distance traveled by Train 1 is:
.

.
and the distance traveled by Train 2 is:
.

.
What else do we know? For one thing we know that both trains travel for 2 hours. Therefore, we can replace both times and by 2 hrs.
.
In addition, we know that the rate for Train 2 is 20 mph more than the rate for Train 1. So we can write:
.

.
Let's make these two substitutions. First, we substitute 2 for both and and our two distance equations become:
.
and
.
Then in the equation for we substitute for and the equation becomes:
.

.
If we then do the distributed multiplication on the right side of this equation by multiplying 2 times each of the quantities in the parentheses, we see that:
.

.
So we now have the two equations:
.
and
.
Finally, what else do we know? Since the trains are going in exactly opposite directions, the distance between them can always be found by adding their distances. So, we can add the two left sides of these equations and similarly add the two right sides to get the equation:
.

.
But we know that the left side of this equation equals 280 miles, the total distance between the two trains. So we replace the left side with 280 and the equation then becomes:
.

.
On the right side of this equation we add the two terms containing to get:
.

.
Next we get rid of the 40 on the right side by subtracting 40 from both sides as follows:
.

.
and by dividing both sides by 4 we find that:
.

.
We now know that the rate for the train going north (we called it ) is 60 mph. And since we know from the given problem that the rate of the train going south is 20 mph faster, we know that the southbound train is going at the rate of 80 mph.
.
Let's check this, just to help ensure that we didn't make a mistake. At 60 mph in 2 hours the northbound train goes 120 miles. And at 80 mph in the same 2 hours the southbound train goes 160 miles. So the distance between the trains is 120 + 160 = 280 miles, just as the problem says it should be. So, our answers are correct.
.
Hope this helps you with the rest of your problems. Just remember: Distance equals Rate times Time. If you have further questions, just post them and hopefully one of the tutors will be able to give you the assistance you need.
.
Good luck! (And consider getting a louder, more obnoxious sounding alarm clock ... LOL)

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In order to be parallel, the graphs of each of these two equations must have the same slope.
.
To find the slopes, let's convert each equation to the slope intercept form. That form is:
.
y = mx + b
.
and m, which is the multiplier of x, is the slope of the graph. (b is the value at which the graph crosses the y-axis.)
.
So let's look at the first equation and let's solve it for y.
.
Start with:
.
3x + 4y = 4
.
Get rid of the 3x on the left side by subtracting 3x from both sides. With that subtraction the equation becomes:
.
4y = -3x + 4
.
Now solve for y by dividing both sides of this equation (all terms) by 4. This results in:
.
y = (-3/4)x + 4/4
.
and the 4/4 results in 1, so the equation becomes:
.
y = (-3/4)x + 1
.
By comparing this to the slope intercept form you can see that m, the slope, is the multiplier of x and it is -3/4.
.
Now, lets do the same thing for the second equation.
.
Start with:
.
2x - 6y = 7
.
Get rid of the 2x on the left side by subtracting 2x from both sides to get:
.
-6y = -2x + 7
.
Solve for y by dividing both sides (all terms) by -6 and the equation becomes:
.
y = (-2/-6)x + 7/(-6)
.
This time the multiplier of the x is (-2/-6) which reduces to +1/3. ( b is equal to -7/6)
.
So for one equation we have the slope equal to -3/2 and for the other equation the slope is +1/3. The two slopes are not the same. Therefore, the graphs have to cross somewhere. And since they cross at some point they are not parallel.
.
You now have the answer and there is nothing else that you need to do.
.
Hope this helps you to understand the problem.
.

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This is the Fibonacci sequence. The first two terms in the sequence are, by definition, 0 and 1. After that, each added term is the sum of the two terms that immediately precede it. An extension of this sequence is:
.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, ...
.
So the answer to this problem is 21 because it is the sum of the 8 and the 13 that are the two terms that come immediately before it.
.
For an explanation of this sequence, you can go to http://en.wikipedia.org/wiki/Fibonacci_number.
.
Hopefully this helps you to understand the Fibonacci number pattern and how we determined the answer to this problem.
.

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You start by choosing 1.
.
From then on, whatever the second player chooses, your next number should be such that when you add it to the second player's choice, the total is 9.
.
For example, you start by choosing 1. Then the second player chooses 2. You then choose 7 because 2 + 7 equals 9. Next the second player chooses 5. You then choose 4 because 5 + 4 equals 9. Then second player chooses 8. You then choose 1 because 8 + 1 equals 9. And so on.
.
Think about this. You began with 1. After that every two choices (his and then yours) adds 9 more. So the game will score as follows:
.
1 + 9 + 9 + 9 + 9 + 9 + 9 = 55
.
This means that after you make the last selection in this series, the total will be 55. After that, whatever number your opponent chooses (1 through 8) will make the total be from 56 to 63 and then it is your turn and you pick the appropriate number to make the total equal 64. You always win.
.
Hope this helps you out.

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Let N be the unknown number.
.
"Increased by 2 thirds of the number" means find 2 thirds of N and add it to (or increase) N by it.
.
So we have N and increase it by 2 thirds N, and this translates to:
.
N + (2/3)N
.
According to the problem, this equals 35. So we have an equation:
.
N + (2/3)N = 35
.
We can get rid of the denominator 3 by multiplying both sides (all terms) by 3 and we then have:
.
3N + 2N = 105
.
Adding together both terms on the left side reduces the equation to:
.
5N = 105
.
Solve for N by dividing both sides of this equation by 5 to get:
.
N = 105/5 = 21
.
That's the answer. The number is 21. 1 third of 21 is 7 so 2 thirds of 21 is 14. When you increase the number (21) by 2 thirds of 21 it is:
.
21 + 14
and this equals 35, just as the problem says it should.
.
Hope this helps you to understand this problem and how you can solve it.
.

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There is a conventional form for a linear equation called the slope-intercept form. The form of this equation is:
.
y = m*x + b
.
and in this equation the m which is the multiplier of x is the slope of the line that is the graph of the equation. And the b is a constant that is the value on the y-axis where the line graph crosses the y-axis.
.
So let's start with the given equation y + 3x = 6 and see if we can't get it into the same form as the slope-intercept form. We do this as follows:
.
Start with the given equation:
.
y + 3*x = 6
.
Move the +3*x to the right side by subtracting 3*x from (or adding negative 3*x to) both sides:
.
y + 3*x - 3*x = -3*x + 6
.
On the left side the 3*x and the -3*x cancel each other and we are left with:
.
y = -3*x + 6
.
If you compare this with the slope-intercept form you can now see that the multiplier of the x is -3, and since the multiplier of the x is the slope, we then know that the slope of the line graph is -3. And in addition, we know that the line graph crosses the y-axis where y equals +6.
.
Hope this helps you to understand this problem a little better. The slope-intercept form of the equation is a very useful form to work with in graphing.
.

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You have two unknowns ... the number of nickels and the number of dimes. Call the unknown number of nickels N and call the unknown number of dimes D.
.
Since there are two unknowns, you will need to have two independent equations to solve this problem. So we will need to look for two equations that we can develop from the information that is given in the problem.
.
The first bit of information is that Sam has 6 more nickels that dimes. That means that if we subtract the number of dimes (D) from the number of nickels (N), the answer is 6. So we can write one equation as:
.
N - D = 6 <--- this is our first equation
.
Next you are told that the total value of the coins is $2.10 which is 210 cents. Since each nickel is 5 cents, if we multiply the number of nickels (N) times 5 we will then know the number of cents we have from nickels. Similarly, each dime is 10 cents, so if we multiply the number of dimes (D) times 10 we will then know the number of cents from dimes. And if we then add 5*N and 10*D the total should be 210 cents. In equation form this is:
.
5*N + 10*D = 210 <--- this is our second equation
.
We can solve these two equations by a variable elimination. Let's return to the first equation and get rid of the D on the left side by adding D to both sides as follows:
.
N - D + D = 6 + D
.
On the left side the -D and the +D cancel each other and we are left with:
.
N = 6 + D
.
Now we can go to the second equation and in it replace N with its equal of 6 + D. With this replacement the second equation becomes:
.
5*(6 + D) + 10*D = 210
.
Do the distributed multiplication on the left side by multiplying 5 times both the terms inside the parentheses. When we do that the equation is:
.
30 + 5*D + 10*D = 210
.
Get rid of the 30 on the left side by subtracting 30 from both sides as follows:
.
30 - 30 + 5*D + 10*D = 210 - 30
.
On the left side the 30 - 30 cancel each other and on the right side the subtraction results in 180. So this equation becomes:
.
5*D + 10*D = 180
.
Combine the two terms on the left side and we have:
.
15*D = 180
.
Solve for D by dividing both sides by 15 to get:
.
D = 180/15 = 12
.
So now we know that there are 12 dimes.
.
We can now return to our first equation and replace D with 12 to get:
.
N - 12 = 6
.
Get rid of the 12 on the left side by adding 12 to both sides to get:
.
N - 12 + 12 = 6 + 12
.
The left side reduces to just N and the right side totals 18, making the equation:
.
N = 18
.
So we now know that there are 18 nickels and 12 dimes.
.
We can see that there are 6 more nickels than dimes. That checks with what the problem says. And we can figure out the total value of the money. Since there are 5 cents for every nickel, we can multiply 5 times the 18 nickels and see that we have 90 cents worth of nickels. And since there are 10 cents for each dime and we have 12 dimes, we multiply 10 times 12 and see that there are 120 cents worth of cents in our 12 dimes. Adding together the 90 cents worth of nickels and 120 cents worth of dimes, we get a total value of 210 cents or $2.10, just as the problem says it should be. So our answer checks out entirely.
.
And we can answer the problem by stating that there are 18 nickels.
.
Hope that this helps you to understand this problem and how to go about solving it.
.

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An arithmetic sequence has a constant difference between each of its terms. If we call this difference D, and the first term is 6, then the second term is 6 + D and the third term is the second term which is 6 + D, plus D. So the third term is 6 + 2D. Similarly the fourth term is the third term which is 6 + 2D plus D. Therefore, the fourth term is 6 + 3D.
.
There is a pattern here. Notice that each term is 6 plus D times one less than the number of the term we are trying to find. So we can see that the twentieth term would be 6 + 19*D.
.
And we are told that the twentieth term equals 63. Therefore, we can write the equation:
.
6 + 19*D = 63
.
From this equation we can find the value of D, the constant difference between terms. First subtract 6 from both sides of the equation to get:
.
19*D = 57
.
Solve for D by dividing both sides of this equation by 19. As a result of this division the equation becomes:
.
D = 57/19 = 3
.
So now we know that the difference between terms is 3. From the problem we were told that the first term is 6. Then the second term is 6 + 3 or 9. Then the third term is the second term plus 3, which means that it is 9 + 3 = 12. And the fourth term is 12 + 3 = 15. And so on ...
.
The answer is that the first three terms of the sequence are 6, 9, and 12.
.
Hope that this helps you to understand a little bit more about arithmetic sequences.
.

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Let x represent the number of pounds of $6 coffee to be added to form a mix of the two coffees.
.
The value added to the mixture by the $6 coffee will be $6 times the number of pounds of the $6 coffee. This value can be written as 6*x.
.
The problem tells you that there are 3 pounds of coffee costing $4 per pound. Therefore, the value of the $4 coffee in the mixture will be 3 pounds times $4 per pound. This product is $12.
.
So the total value of the coffee will be 6*x + $12
.
The number of pounds of coffee in the mixture will be the sum of the number of pounds of the two types of coffee. The problem tells you that there are 3 pounds of the cheaper coffee, and we plan to add x pounds of the more expensive coffee. So when we are done mixing the two types of coffee, the total weight of the mixture will be 3 + x pounds.
.
Then the cost per pound of the mixture will be the total value of the coffee in the mixture divided by the total weight of the mixture. This will be the value 6x + 12 divided by the total weight 3 + x. The problem tells us that this cost per pound is to equal $4.50. Therefore, we can write the equation:
.

.
To get rid of the denominator, multiply both sides by 3 + x as follows:
.

.
On the left side the 3 + x in the numerator cancels with the denominator. On the right side the 4.50 multiplies each of the terms in the parentheses. The result is:
.

.
and this simplifies to:
.

.
Get rid of the 4.50*x on the right side by subtracting 4.50*x from both sides. Then get rid of the 12 on the left side by subtracting 12 from both sides. These two subtractions will result in:
.

.
This simplifies to:
.

.
Finally we can solve for x by dividing both sides by 1.50 and we get:
.

.
This tells us that we should add 1 pound of coffee costing $6 per pound to the 3 pounds of coffee costing $4 per pound and we will get a mixture that should sell for $4.50 per pound.
.
As a check, we can see that the mixture will be a total of 4 pounds. The total value of the coffee will be $6 for the 1 pound of $6 coffee and $12 for the 3 pounds of $4 coffee. Therefore, the total value of the 4 pound mixture is $6 + $12 or $18.
.
The $18 divided by the total weight of 4 pounds results in $4.50 just as the problem wanted. So the answer checks.
.
Hope that this helps you to see how this problem can be solved.
.

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Let x be the bigger number and y be the smaller number.
.
You are told that x (the bigger number) is 6 times y (the smaller number). Therefore we can write the equation:
.
x = 6y
.
Then you are told that the difference between the two numbers is 12. That means that if you subtract y from x, the answer is 12. In equation form this is:
.
x - y = 12
.
But from the first equation, you know that x is equal to 6y. Therefore, in the second equation you can replace x with its equal 6y and this changes the second equation to:
.
6y - y = 12
.
Do the subtraction on the left side and you have:
.
5y = 12
.
Solve for y by dividing both sides of this equation by 5 and you get:
.
y = 12/5
.
And when you divide 5 into 12 you get 2.4 So you know that y, which is the smaller number, equals 2.4 And x, the larger number, equals 6 times y or 6 times 2.4 Multiply 6 times 2.4 and you get 14.4
.
The two numbers are 14.4 and 2.4
.
Check by subtracting the two numbers and you find that the difference is 12, just as the problem says it should be. And when you multiply 6 times the smaller number (6 times 2.4) you get 14.4 So the larger number (14.4) is 6 times the smaller number (2.4) which is also just as the problem says it should be.
.
I hope this helps you to understand the problem a little better and that you can understand a process showing how the problem can be solved.
.

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It may be obvious to you, but let's establish the rule that a tricycle has 3 wheels and a wagon has 4.
.
Let's also establish that T represents the number of tricycles and W represents the number of wagons. If you multiply T by 3 that will be the number of wheels you have due to tricycles, and if you multiply W by 4 that will be the number of wheels due to wagons.
.
The problem tells you that the total number of wheels is 36.
.
Therefore we can write the equation:
.
3T + 4W = 36 (the number of tricycle wheels plus the number of wagon wheels = 36)
.
Let's recognize that the maximum number of wagons is 9. How do we know that? Because if you have 9 wagons, then you have 9 times 4 = 36 and all 36 wheels are on wagons. If you have 10 wagons, then you have 40 wagon wheels and that's not allowed. The maximum number of wheels is 36.
.
Similarly, the maximum number of tricycles is 12. We know that because 12 tricycles will have a total of 36 wheels, the maximum allowed.
.
Now let's solve our equation for one of the variables in terms of the other. Let's say we solve for T in terms of W by doing the following:
.
Start with:
.
3T + 4W = 36
.
Subtract 4W from both sides to get:
.
3T = 36 - 4W
.
Solve for T by dividing both sides (all terms) by 3 to get:
.
T = 12 - (4W)/3
.
Now we can just assume values for the W, the number of wagons and see what this results in for T. Remember, we decided that there are at most 9 wagons, so we need to check for values of W equal to 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. All we need to do is substitute those values into our equation and see which of them result in a whole number for the number of tricycles (not a mixed number involving a fractional part of a tricycle. That wouldn't make sense.)
.
So let's start:
.
If W = 0 then (4W)/3 = 0 & T will be equal to 12 - 0 = 12. That works.
If W = 1 then (4W)/3 = 4/3 & T will be equal to 12 - 4/3 = 10&2/3. Won't work.
If W = 2 then (4W)/3 = 8/3 & T will be equal to 12 - 8/3 = 9&1/3. Won't work.
If W = 3 then (4W)/3 = 12/3 & T will be equal to 12 - 12/3 = 8. That works.
If W = 4 then (4W)/3 = 16/3 & T will be equal to 12 - 16/3 = 6&2/3. Won't work.
If W = 5 then (4W)/3 = 20/3 & T will be equal to 12 - 20/3 = 5&1/3. Won't work.
If W = 6 then (4W)/3 = 24/3 & T will be equal to 12 - 24/3 = 4. That works.
If W = 7 then (4W)/3 = 28/3 & T will be equal to 12 - 28/3 = 2&2/3. Won't work.
If W = 8 then (4W)/3 = 32/3 & T will be equal to 12 - 32/3 = 1&1/3. Won't work.
and finally,
If W = 9 then (4W)/3 = 36/3 & T will be equal to 12 - 36/3 = 0. That works.
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So we found that the possible combinations are:
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zero wagons and 12 tricycles
3 wagons and 8 tricycles
6 wagons and 4 tricycles
9 wagons and zero tricycles
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Each of these four possibilities will result in a total of 36 wheels.
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You could probably reason your way through this problem as fast as developing the equation as above. First recognize that the maximum number of tricycles is 12 because that would account for all 36 wheels. Then recognize that the number of tricycles must be an even amount because an odd number of tricycles will create an odd number of wheels and when you subtract an odd number of wheels from 36 you would end up with an odd number of wheels left for wagons. But wagons require an even number of wheels.
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So the possible numbers of tricycles would be 0, 2, 4, 6, 8, 10, and 12. Then all you have to do is for each of these 7 numbers, multiply by 3 wheels, subtract the answer from 36 and see if that result is exactly divisible by 4. If it is, then it is one of the answers.
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For example, let's try 2 tricycles. That means 3 times 2 or 6 wheels for tricycles. Subtract those 6 wheels from 36 and you have 30 wheels left for wagons. But that answer, 30 wheels, is not exactly divisible by 4, so you don't get a whole number of wagons. Therefore, 2 tricycles is not a good number.
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Now let's try 4 tricycles. That means 12 wheels for tricycles and that leaves 24 wheels for wagons. 24 wagon wheels is 6 wagons exactly, and so 4 tricycles and 6 wagons gives you 36 wheels.
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You can work out the remaining 5 numbers for the possible number of tricycles and you will see that you get the same four possible solutions as we did with the equation.
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I hope this helps you to understand the problem you were having difficulty with.
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Let's call the smaller number S. And let's call the larger number L.
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The problem tells you that the sum of the smaller number, S, and the larger number, L, equals 30. So we can write the equation:
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S + L = 30
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Next the problem says that the larger number (L) is twelve more than one half the smaller number (S).
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One half the smaller number is S divided by 2 which is S/2. 12 more than that is:
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S/2 + 12
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and this equals the larger number. So we can write another equation:
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S/2 + 12 = L
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Now we have two independent equations so we can solve for the two unknowns. The equations are:
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S + L = 30 and
S/2 + 12 = L
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Since the second equation says that L = S/2 + 12, we can substitute the right hand side of this equation for the L that appears in the first equation we wrote above. Making this substitution makes the first equation become:
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S + S/2 + 12 = 30
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Eliminate the +12 on the left side by subtracting 12 from both sides to get:
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S + S/2 = 18
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Next, add the two terms on the left side as follows:
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S + S/2 = 2S/2 + S/2 = 3S/2
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(Note that S = 2S/2 gives both terms the common denominator of 2, so the two terms can be added by adding their numerators and putting that result over the common denominator 2.)
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Replace the S + S/2 with its equivalent 3S/2 and the equation becomes:
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3S/2 = 18
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Get rid of the denominator 2 by multiplying both sides of this equation by 2 to get:
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3S = 36
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Solve for S by dividing both sides by 3 and you have:
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S = 12
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Now we know that the smaller number is the even number 12. And since the sum of the two numbers is 30 we also know that the larger number must be 18 (because 18 + 12 = 30) which is also an even number.
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You can check this problem by ensuring that the larger number (18) is 12 more than half the smaller (half of 12). Since 18 is 12 more than 6, this checks. Therefore, we know that this answer is correct ... namely the smaller even number is 12 and the larger even number is 18.
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Hope this explanation helps you to understand the problem a little better.
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If the smaller integer is N, then the next consecutive integer is N+1.
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If you add the smaller to three times the larger, this can be written as:
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N + 3*(N+1)
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and the problem tells you that this sum equals 159. Therefore, we can write the equation:
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N + 3*(N+1) = 159
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Do the distributed multiplication on the left side by multiplying 3 times each of the terms in the parentheses to get:
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N + 3N + 3 = 159
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Add the two N terms to get:
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4N + 3 = 159
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Eliminate the 3 on the left side by subtracting 3 from both sides:
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4N = 156
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Solve for N by dividing both sides by 4 and you get:
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N = 39
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You were asked by the problem to solve for the smaller integer, and we called that integer N. So you have the answer ... the smaller integer is 39 and this means that the larger integer is 40.
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Check by adding the smaller (39) to 3 times 40. This means add 39 to 120 and you get 159, just as the problem said it should be.
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Hope this helps you to understand the problem and how to solve it.
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Let's assume that theta equals 90 degrees.
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If theta equals 90 degrees, then (1/2)*(theta) is equal to 45 degrees, and sin(45) is approximately 0.7071.
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Next, what is sin(theta) for theta = 90 degrees?. The answer to that is +1. This means that if theta equals 90 degrees, then:
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(1/2)* sin(theta) = (1/2)*(+1) = 1/2 = 0.5000
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Therefore, we can say that if theta equals 90 degrees then:
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sin((1/2)*theta) = 0.7071 and
1/2*sin(theta) = 0.5000
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With this example, you can now say that it is not always true that:
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sin(1/2(theta))=1/2sin(theta)
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Hope this helps you to understand a way this problem can be answered and what the answer is.
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First, determine how many possible pairings can be formed using the three persons. The possible pairings are Werner and Myrna, Werner and Virna, and Virna and Myrna.
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One of these pairings must be the two persons who are telling the truth. So let's examine the statements each person makes in the pairings.
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In the pairing of Werner and Myrna: Werner says, "Virna was born first" and Myrna says, "Werner is the oldest." There is a conflict in these two statements, so one of them has to be lying.
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In the pairing of Werner and Virna: Werner says, "Virna was born first" and Virna says, "I am not the oldest." There is a conflict in these two statements, so one of them has to be lying.
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In the pairing of Virna and Myrna: Virna says, "I am not the oldest" and Myrna says "Werner is the oldest." There is no conflict in what these two are saying, so they are both telling the truth. Therefore, Myrna is correct when she says, "Werner is the oldest." So Werner was born first, and Werner is also the liar.
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Hope this helps you to see a way of thinking your way through this puzzle.

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You are very close to having the correct answer.
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Do the problem like this. You already know that you need to get 450 total points on the tests.
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So far you have scored a total of 263 points. (You determine this by adding 87, 92, and 84 points.)
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Subtract the 263 points you already have from the 450 points that you need to get in total and you find out that on the last test you need to get 187 points. Divide that by 2 and you find that you need to score 93.5 on the test. Since the test counts double, a score of 93.5 will double to give you the 187 points that you need. If you cannot get half-points on the last test, you will need a score of 94 (doubling to 188 points and resulting in 451 total points) to ensure that you get a 90 overall average.
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I hope this helps you to understand the problem.
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