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# Recent problems solved by 'bucky'

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 Inequalities/116349: The lowest temperature ever recorded in the state of Oregon was -54F (in Seneca, on Feb 10, 1933) The state's record high temperature occurred in Pendleton on Aug 10, 1898, when it reached 119F. What is the historical temperature range in the state of Oregon?1 solutions Answer 84623 by bucky(2189)   on 2007-12-12 15:58:23 (Show Source): You can put this solution on YOUR website!I think this is what the problem is looking for you to do: . At the low end the temperature has always been equal to or greater than -54 degrees F. . At the high end the temperature has always been equal to or less than 119 degrees F. . So the temperature has always been in the range from -54 to +119 degrees Fahrenheit (inclusive -- meaning that the range includes -54 and +119). . You can express this in the form of a trichotomy inequality: . . where the numbers are understood to be in units of degrees Fahrenheit and this inequality says that the temperature has always been between the two limits ... bigger than or equal to -54 and less than or equal to +119. . Hope this helps you to understand the concept of range. .
 Radicals/116380: radical 50 + radical 181 solutions Answer 84619 by bucky(2189)   on 2007-12-12 15:42:33 (Show Source): You can put this solution on YOUR website!You are given: . . To simplify this you can use the rule that says: . . Applying that rule to we can break the 50 up into the product of 25*2. So we can say that . But . So we can substitute 5 for and the problem simplifies further: . . Now let's do the same thing for . Factor 18 so that you have: . . And so this part of the problem simplifies further to: . . Put the two parts of the problem together ... . . If you factor from the two terms on the right side you get: . . So the answer to this problem is: . . Hope this helps you to understand one of the rules that can be used in simplifying radicals. .
 percentage/116377: Find the percent of the number 13. On a math test, you obatained a score of 66 points. This was 88% of the total number of points on the test . How many total points were on the test? Any help with this equation would be Greatly appreciated.1 solutions Answer 84618 by bucky(2189)   on 2007-12-12 15:25:15 (Show Source): You can put this solution on YOUR website!Let X represent the total number of points on the test. . The problem tells you that 88% of X equals 66 points . Change 88% to its equivalent decimal form of 0.88. . Then multiply 0.88 times X and set that equal to 66 points. In equation form this is: . 0.88X = 66 . Solve for X by dividing both sides of this equation by 0.88 and you get: . . Calculator time ... when you divide 66 by 0.88 you get 75. So the answer to this problem is: . . There are 75 points on the exam. . Check: If the test has 75 points and you got 88% (or 0.88) of the points correct then you scored 0.88 times 75 which equals 66 points. This checks with the information in the problem. . Hope this helps you to understand percent a little more. .
 Linear-equations/116327: Hello, can you help me solve this graphing problem? I need to find the x and y intercepts and then graph -x+4y=4 and y = -4x-4.How do I go about finding out the intercepts, what do they want me to do? This is from my homework and is not in a book.Can you help me please and explain step by step? Thanks chama1 solutions Answer 84589 by bucky(2189)   on 2007-12-12 11:46:25 (Show Source): You can put this solution on YOUR website!Chama --- . You probably can just about do these problems in your head once you understand what is going on. . Begin by thinking about the coordinate system of axes. A point on the y-axis must have a corresponding x value of zero. For example, what is the coordinate pair that corresponds to the point +5 on the y-axis. The pair is (0, +5). That means that for a given linear function if you set x equal to zero, y will equal the intercept value on the y-axis. . Similarly, any point on the x-axis will have zero for its y value. Example, the point (-3, 0) is on the x-axis. So if you set y equal to zero in the given function, the corresponding value of x will be on the x-axis. . Think about that until you understand the basic concept. Now let's do the problems. . First problem ... given -x + 4y = 4 . Find the y-intercept by setting x equal to zero and you get 4y = 4. Solve for y by dividing both sides by 4 to get y = 1. So the graph crosses the y-axis at +1. . Next, find the x-intercept by setting y = 0 and you get -x = 4 which becomes x = -4. So the graph crosses the x-axis at -4. . You can graph this line by plotting the x-intercept point on the x-axis and the y-intercept point on the y-axis and the drawing a straight line running through these two points. The graph of -x + 4y = 4 shows these values with the graphed line running through them: . . Second problem. Given: y = -4x -4 . Set x equal to zero and the function becomes y = -4. So the y-intercept is -4. . Then set y = 0 and you get 0 = -4x - 4. Add 4 to both sides and you have 4 = -4x. Divide both sides by -4 and you have -1 = x. So the x-intercept is at -1. . The graph is: . . and it shows the intercepts we found. . Hope this helps you to understand the problems and how to solve them. .
 Linear-equations/116283: Can you help me with these questions? I did not get this out of a book, it was a handout. The question is: Find the x and y intercepts and then graph x-y=2 , -x+4y=4 and y = -4x -4. I do not fully understand the question. Any help on this is greatly appreciated. Can you explain it step by step, not leaving anything out , please. Thankyou Laurie1 solutions Answer 84588 by bucky(2189)   on 2007-12-12 11:21:22 (Show Source): You can put this solution on YOUR website!Laurie --- . You probably can just about do these problems in your head once you understand what is going on. . Begin by thinking about the coordinate system of axes. A point on the y-axis must have a corresponding x value of zero. For example, what is the coordinate pair that corresponds to the point +5 on the y-axis. The pair is (0, +5). That means that for a given linear function if you set x equal to zero, y will equal the intercept value on the y-axis. . Similarly, any point on the x-axis will have zero for its y value. Example, the point (-3, 0) is on the x-axis. So if you set y equal to zero in the given function, the corresponding value of x will be on the x-axis. . Think about that until you understand the basic concept. Now let's do the problems. . Given: x - y = 2 . If x is zero and you solve for y you get -y = 2 which becomes y = -2. So the point (0, -2) is the y-intercept. Then if y is zero the function becomes x = +2. So the point (2, 0) on the x-axis is the x-intercept. . You can get the graph for each of these three problems by plotting the intercepts on the x and y axes and drawing a straight line extending through them. The graph for the first function ... x - y = 2 ... is: . . and this shows the intercept points that we found. . Next problem ... given -x + 4y = 4 . Find the y-intercept by setting x equal to zero and you get 4y = 4. Solve for y by dividing both sides by 4 to get y = 1. So the graph crosses the y-axis at +1. . Next, find the x-intercept by setting y = 0 and you get -x = 4 which becomes x = -4. So the graph crosses the x-axis at -4. The graph of -x + 4y = 4 shows these values with the graphed line running through them: . . Final problem. Given: y = -4x -4 . Set x equal to zero and the function becomes y = -4. So the y-intercept is -4. . Then set y = 0 and you get 0 = -4x - 4. Add 4 to both sides and you have 4 = -4x. Divide both sides by -4 and you have -1 = x. So the x-intercept is at -1. . The graph is: . . and it shows the intercepts we found. . Hope this helps you to understand the problems and how to solve them. . Bucky
 Permutations/116297: Simplify: (-11)(-5)1 solutions Answer 84582 by bucky(2189)   on 2007-12-12 10:48:22 (Show Source): You can put this solution on YOUR website!The algebraic rule is that if you multiply two numbers together and they both have the same sign, the answer is positive. . If you multiply two numbers together and they have different signs (one number is positive and the other is negative) the answer is negative. . In this problem, both numbers have the same sign ... both are negative. Therefore, the answer must be positive. Multiply the 5 times the ll and get 55 and make it a positive answer or +55. . Hope this helps you to understand algebraic multiplication a little better. .
 expressions/116299: Simplify: 5(2a + b) - (a + 3b)1 solutions Answer 84568 by bucky(2189)   on 2007-12-12 08:26:09 (Show Source): You can put this solution on YOUR website!Given: . 5(2a + b) - (a + 3b) . Do the distributed multiplication by multiplying 5 times each of the two terms in the first set of parentheses. When you do you get: . 10a + 5b - (a + 3b) . The second set of parentheses is preceded by a minus sign. Because of the minus sign if you remove the parentheses you need to change the signs of the terms inside the parentheses. When you remove these parentheses the expression becomes: . 10a + 5b - a - 3b . Now, combine the like terms. 10a combines with -a to give 9a. And 5b - 3b combines to give +2b. As a result, the entire expression is simplified to: . 9a + 2b . Hope this helps you to understand the problem ... and some techniques for simplifying algebraic expressions. .
 Linear-equations/116228: #1 Write the equation of a line in y= form, through the point (-2,5) and (3,4). I have found the first part of the problem by doing y2-y1 over x2-x1 and that gave me -1/5. But I do not understand how to do the rest of the problem and write out the answer. #2 Find the slope and y-intercepts of the given line: 4x-5y=16. What steps do I take to solve this problem?1 solutions Answer 84553 by bucky(2189)   on 2007-12-12 00:26:09 (Show Source): You can put this solution on YOUR website!One equation you can take advantage of is the form: . . This is called the slope-intercept form and it is the equation of a line in which m, the multiplier of the x term, is the slope of the line and b is the value on the y-axis where the line crosses the y-axis. . In the first problem you correctly found that the slope was -1/5. You can substitute this value into the slope-intercept form and you get that the equation for the line then becomes: . . At this point you just need to find b to complete the equation. To do this you can use either of the two points that you were given. Let's use the point (3, 4). This tells you that when x = 3 and y = 4 the equation must balance. So substitute 3 for x and 4 for y and the equation becomes: . . Multiply out the first time on the right side to get: . . Solve for b by adding +3/5 to both sides. [This gets rid of the on the right side.] On the left side when you add you get . This results in b = . So we can substitute this value for b into the slope-intercept form to get: . . Let's check it out by finding if (-2,5) is a point on the line. Substitute into the slope-intercept form -2 for x and 5 for y and see if the equation balances. . . Replace 5 by its equivalent and this becomes: . . Multiply out the first term on the right side: . . The two terms on the right side add to and so the equation is true. This checks the equation . The second problem asks you to find the slope and y-intercept of the equation: . . Let's work this into the form . Begin by subtracting 4x from both sides to get rid of the 4x on the left side. This changes the equation to: . . Make the term on the left side positive by multiplying both sides (all terms) by -1 to get: . . Solve for y by dividing both sides (all terms) by 5 to get: . . Note that this is in the slope-intercept form. So just by looking at it you can tell that the slope is [you can tell that because it is the multiplier of x which is the slope.] . And you can tell that the y-intercept is because that is the constant term in this slope-intercept equation. . Hope this helps you to understand these problems. Since it's late, please check my math to make sure I didn't make a dumb error in the numbers. The basic process of using the slope- intercept form is correct. .
 Graphs/116265: Solve by graphing: x^2+y^2=25 and x-2y=-5 I graphed my circle and line perfectly but when it came to using either addition or substitution methods to find the solution set (where the line crosses the circle at two points) I had problems:( I can see by looking at my graph that one solution is (-5,0) and the other is approximately (3.5, 4.5) but I have to use one of the above methods to show how I came up with my solutions and show my work. This is what I've done so far: Using substitution method x^2+y^2=25 and x-2y=-5 (changed to x=2y-5) and substituted in to 1st equation (2y-5)^2+y^2=25 4y^2-20y+25+y^2=25 (combine like terms) 5y^2-20y+25=25 (subtract 25 from each side) 5y^2-20y=0 This is where I loose it, because in class all of our problems worked out nice and neat and all I had to do was factor my quadratic equation to come up with one solution and then plug that into one of my original problems to get my second. No one in study group tonight quite got this one either and this is due this Thursday!1 solutions Answer 84545 by bucky(2189)   on 2007-12-11 23:24:56 (Show Source): You can put this solution on YOUR website!Good Job ... . x^2+y^2=25 and x-2y=-5 (changed to x=2y-5) and substituted in to 1st equation (2y-5)^2+y^2=25 <=== OK . 4y^2-20y+25+y^2=25 <=== OK . (combine like terms) 5y^2-20y+25=25 <=== OK . (subtract 25 from each side) 5y^2-20y=0 <==== . Now just factor 5y out of the two terms on the left side to get: . 5y*(y - 4) = 0 . This equation will be true if either of the two factors on the left side equals zero because a multiplication by zero on the left side makes the left side zero and, therefore, equal to the zero on the right side. . So there are two possible answers for y (because the line crosses the circle at two points). . Set the factors equal to zero ... . First: . 5y = 0 ... divide both sides by 5 to get y = 0 . Next: . y - 4 = 0 ... add 4 to both sides ... . y = 4 . Plug these two values for y (0 and 4) into the equation x - 2y = -5. [Of the two equations this will probably be the easier one to work with.] . When y = 0 the equation x - 2y = -5 becomes x = -5. So (-5, 0) is one of the intersection points. . When y = 4 the equation x - 2y = -5 becomes x - (2*4) = -5 which simplifies to x - 8 = -5 . Add 8 to both sides and you get x = +3. So the point (3, 4) is the second point at which the line crosses the circle. . You did a very good job on the hardest part of the problem ... you just needed a little hint at the very last part. Keep up the good work ... good luck to you and your study group. .
 Polynomials-and-rational-expressions/116144: Can someone please help me figure out how to do this problem. I have tried many times and I just keep getting more confused. Thank you in advance! Multiply the fraction. 2x squared - x - 3 OVER 3x squared + 7x + 4 TIMES 3x squared - 11x - 20 OVER 4x squared - 91 solutions Answer 84540 by bucky(2189)   on 2007-12-11 22:49:06 (Show Source): You can put this solution on YOUR website!Given: . . If you factor all these polynomials, you get: . . Cancel terms in one of the denominators with the same term in one of the numerators as follows: . . You are left with the answer: . . Hope this helps you to find where you are having trouble. .
 Numbers_Word_Problems/116140: . A solid metal cube of side 6 inches is placed in a rectangular tank whose length, width, and height are 1, 2, and 3 feet. What is the volume, in cubic units, of water that the tank can now hold? 1 solutions Answer 84438 by bucky(2189)   on 2007-12-11 12:55:11 (Show Source): You can put this solution on YOUR website!This is how I interpret what the problem is asking you to find. . When empty, the tank can hold a maximum of 6 cubic feet of water. You find that volume by multiplying the three dimensions of the tank ... 1 ft times 2 ft times 3 ft = 6 cu ft. . When the tank is empty you put the metal cube in it. The question then is how many cubic feet of water can you then put into the tank before the tank is full. . Since the metal cube is 6 inches on a side, each side is foot long. Therefore, the volume of this cube is the product of or 1/8 cubic ft . Once the cube is in the tank, water cannot occupy that 1/8 cubic ft of space. So the amount of remaining space that water can occupy is the original 6 cubic feet less 1/8 cubic ft which is 5 7/8 cubic feet or in decimal form 5.875 cubic feet. . Hope this helps you to understand the problem and how you can work it. .
 percentage/116110: Write the fraction as a percent. 7. 29/100 Any help with this equation would be Greatly appreciated.1 solutions Answer 84435 by bucky(2189)   on 2007-12-11 12:38:09 (Show Source): You can put this solution on YOUR website!Percent is defined as "hundredths". So, if you can get any fraction in the form where it has 100 as its denominator, you can say that the numerator is the percent. . In this case you are given: . as the fraction. . Since the denominator is 100, 29 is the percent. So the answer is 29% . Suppose though that you were given: . . and asked what percent that was. To convert the 25 so that it becomes 100 you multiply it by 4. But if you multiply it by 4 you must also multiply the numerator 3 by 4. When you do those two multiplications you get: . . Now that you have the denominator equal to 100, you can say the percent is the corresponding numerator of 12. This means that is equivalent to and this corresponds to 12% . Hope that helps you understand one way to convert a fraction to percent. .
 Quadratic_Equations/116122: This question is from textbook Introductury Algebra Please can you help me with this equation, I've been working it out but I'm not able to get the right answer. Graph and solve the equation. y= x^2 - 41 solutions Answer 84430 by bucky(2189)   on 2007-12-11 12:20:53 (Show Source): You can put this solution on YOUR website!Sometimes it's easier to just do a little analysis and not get lost in math. Let's try that approach. . First, the right side of the equation is a quadratic form (the highest power of x is the 2nd power). From that you know that the graph will be a parabola. And since the sign on the x-squared term is positive, you can tell that as you move from the left to the right along the x-axis the graph will drop more and more until it reaches some low point, and then it will start to rise more and more as you move further to the right. You also know that if you draw a vertical line through that low point on the graph the graph to the left of that line will look like the mirror image of the graph to the right of that line. . [For your information in other problems of this type, if the sign on the x-squared term is negative, you can tell that the graph will rise to a peak and then fall back down the further you go to the right along the x-axis.] . Now you can just calculate some points to help you understand what the graph looks like. . One nice and easy point on the graph can be found by setting x equal to zero and finding what the corresponding value of y is. When you set x equal to zero in the given equation for y, you get: . y = (0)^2 - 4 = -4 . Now you know that the point (0, -4) is on the graph. Plot it. . Next let's try letting x = +1 and calculate y. Substituting x into the given equation results in: . y = (1)^2 - 4 = 1 - 4 = -3 . This time you know that the point (1, -3) is on the graph . Next let's set x equal to -1 and calculate the corresponding value of y. . y = (-1)^2 - 4 = 1 - 4 = -3 . Now you know that (-1, -3) is on the graph. . Now let's set x = 2 to find that: . y = (2)^2 - 4 = 4 - 4 = 0 . This time you know that (2, 0) is on the graph. Then if you set x = -2 you get the corresponding value of y to be: . y = (-2)^2 - 4 = +4 - 4 = 0 . The point (-2, 0) is on the graph. . If you have plotted these five points on the graph you should have a pretty good idea of what the parabola looks like. You can use more values for x to find corresponding values of y and get more points on the graph. With more points you should find that it looks like: . . If you are to "solve" this equation that means that you are looking for the value of x where the graph crosses the x-axis. If you've done your graphing pretty accurately you should see that the graph crosses the x-axis at two places ... where x = -2 and x = +2. This means that for those two values of x the corresponding value of y is zero. . Hope this helps you with a little different slant on this problem. .
 Problems-with-consecutive-odd-even-integers/116119: Can you tell me if this is correct? Find three consecutive integers such that four times the sum of the first and second is 17 more than 7 times the third 1st integer:n 2nd integer:n+2 3rd integer:n+4 4n+7(n+4)+171 solutions Answer 84423 by bucky(2189)   on 2007-12-11 11:45:19 (Show Source): You can put this solution on YOUR website!What you have done is listed the integers as being two numbers apart ... consecutive even integers (such as 10, 12, 14, ...) are two numbers apart. So are consecutive odd integers (such as 17, 19, 21 ...). But consecutive integers ... such as 11, 12, and 13 ... are just one apart. Therefore, you represent three consecutive integers as: . 1st integer: n 2nd integer: n + 1 3rd integer: n + 2 . The sum of the first and second is n + n + 1 = 2n + 1 and four times that is: . 4(2n + 1) . Seven times the third is 7(n + 2) . The problem says that 4(2n + 1) is 17 more than 7(n + 2). So if you take 17 away from 4(2n + 1) it will equal 7(n + 2). In equation form this is: . 4(2n + 1) - 17 = 7(n + 2) . Multiply out the two distributed multiplications. First multiply 4 times each of the two terms in its related parentheses. Then multiply 7 times each of the two terms in its related parentheses. When you do those multiplications the equation becomes: . 8n + 4 - 17 = 7n + 14 . Combine the +4 and -17 on the left side and you have: . 8n - 13 = 7n + 14 . Get rid of the -13 on the left side by adding 13 to both sides to reduce the equation to: . 8n = 7n + 27 . Then get rid of the 7n on the right side by subtracting 7n from both sides. When you do that subtraction you get: . n = 27 . If n is 27, then the next two consecutive integers are 28 and 29. So the three consecutive numbers that you are looking for are 27, 28, and 29. . Check: 4 times (27 + 28) = 4(55) = 220. 7 times 29 = 203 Is the difference between 220 and 203 equal to 17. Yes it is, so the answer checks. . Hope this helps you to see where you made your mistake and what you can do to solve the problem. .
 percentage/116105: Write The percent as a fraction in simplest form. 3. 15% Any help with this equation would be Greatly appreciated.1 solutions Answer 84414 by bucky(2189)   on 2007-12-11 09:35:56 (Show Source): You can put this solution on YOUR website!Percent is defined as "hundredths" ... . So in fraction form 15% is converted to . But this fraction can be reduced to a simpler form because both the numerator and denominator can be divided by the common factor of 5. When you divide the numerator 15 by 5 it becomes 3 and when you divide the denominator 100 by 5 it becomes 20. So the reduced fraction is and you can't reduce that fraction any further because 3 and 20 cannot be divided by any number that is common to both of them (other than 1 ... which does not change the problem). . So the answer is 15% = . Hope this helps ...