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# Recent problems solved by 'bucky'

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 Expressions-with-variables/590116: Tell how many solutions the system has. do not actually solve. 5x+4y=-1 25x+20y=-51 solutions Answer 375225 by bucky(2189)   on 2012-03-23 04:29:54 (Show Source): You can put this solution on YOUR website!You are given the system of two equations: . 5x+4y=-1 25x+20y=-5 . 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. . 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: . (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 . (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. . 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: . 5x+4y=-1 25x+20y=-5 . 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: . 25x + 20y = -5 . 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. . 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. .
 Expressions-with-variables/590639: A number plus the reciprocal of the number is 5/4 of the number. What is the number?1 solutions Answer 375221 by bucky(2189)   on 2012-03-23 03:55:04 (Show Source): You can put this solution on YOUR website!Let N represent the unknown number. . By definition the reciprocal of the number is . So the sum of the unknown number and its reciprocal is: . . 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 . So we can write the equation that says this as: . . 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: . . Cross out the denominators with the corresponding terms in the numerator as follows: . . With these cancellations we are left with: . , And doing the multiplications in each of the terms simplifies this equation to: . . 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: . . Doing the subtraction on the right side simplifies this to: . . Now solve for N by taking the square root of both sides: . . After taking the square root of both sides you are left with two answers. Either: . or . 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. . 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: . . or: . . 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. . Hope this helps you to understand the problem a little better. .
 Numbers_Word_Problems/588550: The sum of 2 numbers is 77. Their difference is 7. Find the 2 numbers 1 solutions Answer 374552 by bucky(2189)   on 2012-03-19 04:13:58 (Show Source): You can put this solution on YOUR website!Let X and Y represent the two unknown numbers. Since we have two unknowns, we need two independent equations to solve for them. . The problem tells you that the sum of these two numbers equals 77. So let's write an equation that says that: . X + Y = 77 . The problem also tells you that the difference between the two numbers is 7. Now let's write an equation that says that: . X - Y = 7 . 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: . X + Y = 77 X - Y = 7 . 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: . 2X = 84 . We now solve for X by dividing both sides of this new equation by 2 to get: . X = 42 . 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: . X + Y = 77 . Substitute 42 for X and this becomes: . 42 + Y = 77 . 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: . Y = 77 - 42 = 35 . 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. . Hope this helps you to understand the problem a little more.
 Money_Word_Problems/584664: to use a certain computer data base, the charge is \$30 per hour during the day and \$10.50 per hour at night. if a research company paid \$411 for 28 hours of use, find the number of hours charged at the daytime reate and at the nighttime rate 1 solutions Answer 372868 by bucky(2189)   on 2012-03-09 08:53:33 (Show Source): You can put this solution on YOUR website!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. . 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: . D + N = 28 . 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: . 30D + 10.5N = 411 . 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: . N = 28 - D . 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: . 30D + 10.5*(28 - D) = 411 . 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: . 30D + 294 - 10.5D = 411 . On the left side combine the two terms that contain D by subtracting 10.5D from 30D to get: . 19.5D + 294 = 411 . Next, get rid of the 294 on the left side by subtracting 294 from both sides. The resulting equation is: . 19.5D = 117 . Finally, solve for D by dividing both sides by 19.5 to get: . D = 117/19.5 = 6 . 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. . 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. . The answer therefore is that the computer was used 6 hours during the day and 22 hours at night. . 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. .
 Money_Word_Problems/584166: 28 is less than 12 more than 8 times a number1 solutions Answer 372840 by bucky(2189)   on 2012-03-09 01:44:48 (Show Source): You can put this solution on YOUR website!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. . Then there is the wording "8 times a number" which you can translate to 8 times X or just 8X. . 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 . Finally you are told that 28 is less than that. So you can write the inequality: . 28 < 12 + 8X . 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.) . 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: . 16 < 8X . Finally solve the inequality for X by dividing both sides by +8. The result is: . 2 < X . 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. . Hope this helps you to understand this inequality problem. .
 Equations/584520: I am having trouble with writing out the equations that are in word form. Please help me. -Write an equation and solve: four more than twice a number m is the sum of 4 and 8.1 solutions Answer 372809 by bucky(2189)   on 2012-03-08 20:36:41 (Show Source): You can put this solution on YOUR website!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. . 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. . Putting this all together we can write the equation: . 4 + 2m = 4 + 8 . 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: . 4 - 4 + 2m = 4 - 4 + 8 . 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: . 2m = 8 . At this point you can solve for m by dividing both sides by 2 to get: . m = 8/2 = 4 . So you have written the equation as 4 + 2m = 4 + 8, and you have solved this equation to find that m equals 4. . 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.
 Complex_Numbers/581154: Please show the steps on how to solve this multiplication problem. (3 - squarerootof-5) * (1 + squarerootof-1) (in other words..) quantity (three minus squareroot of negative five) multiplied by quantity (one plus squareroot of negative one) answer is in a + bi form and is as follows... (3 + squarerootof5) + (3 - squarerootof5)i Thank you so much. 1 solutions Answer 371481 by bucky(2189)   on 2012-03-02 04:49:35 (Show Source): You can put this solution on YOUR website!Given to expand: . . Here's a long explanation of this problem that might give you some insight into how you do it. . 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. . 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 . . This means that in the first set of parentheses, the term: . can be replaced by . Furthermore, under the algebraic rules for square roots, this replacement can be split into two parts as follows: . . 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: . . So going back to the original problem, the contents of the first set of parentheses can be written as: . . 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 ] . 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 . . We have now converted the original problem to: . . 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: . . And the multiplication by -1 makes it become: . . Finally, add these two products as follows: . . 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: . . Factor out i from the last two terms and the result is: . . and that's the answer that you said we should get. . 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 .
 Exponents/581121: Could someone please tell me if I am right? 3x^-5/2x^3 Is the answer 3/2x^8 1 solutions Answer 371472 by bucky(2189)   on 2012-03-02 01:26:17 (Show Source): You can put this solution on YOUR website!Given to simplify: . . You got the answer correct if you meant it to be interpreted as: . . 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 . . So the answer to the problem could have been written as: . which is equivalent to . 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: . . 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!!! .
 Travel_Word_Problems/577747: Two trains leave a city at the same time. One travels North and the other travels South 20mph faster, in 2 hours; the trains are 280 miles apart. Find their rates. (I over slept for my class and I HAVE NO IDEA how to do the assigned homework. This should help me figure the rest of the questions out. Thank you thank you thank you!1 solutions Answer 370294 by bucky(2189)   on 2012-02-25 02:43:54 (Show Source): You can put this solution on YOUR website!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)
 Linear-equations/576060: Are the following two lines parallel? 3x + 4y = 4 2x - 6y = 71 solutions Answer 369715 by bucky(2189)   on 2012-02-21 21:58:52 (Show Source): You can put this solution on YOUR website!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. .
 Subset/575139: which is the next number in this logical sequence .. 1,1,2,3,5,8,13, ...1 solutions Answer 369392 by bucky(2189)   on 2012-02-20 06:05:41 (Show Source): You can put this solution on YOUR website!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. .