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# Recent problems solved by 'Theo'

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 Linear_Equations_And_Systems_Word_Problems/467163: Determine which, if any, of the three statements are equivalent. Give a reason for your conclusion. Show complete work and submit your solution to the Dropbox. I) If tomorrow is not Monday, then today is not Sunday. II) Today is not Sunday and tomorrow is not Monday. III) It is not the case that today is Sunday and tomorrow is Monday. a. I and II are equivalent b. II and III are equivalent c. I and III are equivalent d. None are equivalent e. I, II, and III are equivalent (Points : 4) 1 solutions Answer 320454 by Theo(3464)   on 2011-07-05 08:28:38 (Show Source): You can put this solution on YOUR website!statement II and III are equivalent. statement II states: TODAY IS NOT sunday AND TOMORROW IS NOT monday statement III states: IT IS NOT THE CASE THAT (today is sunday and tomorrow is monday). THE CASE is the statement that (today is sunday and tomorrow is monday). IT IS NOT THE CASE THAT negates the statement within the case.
 Expressions-with-variables/467095: Hello, I have a question on solving formulas for variables. What are the steps for solving this problem. x/5 + 5y = m/n; solve for y Thank you1 solutions Answer 320377 by Theo(3464)   on 2011-07-04 17:09:27 (Show Source): You can put this solution on YOUR website!your equation is: x/5 + 5y = m/n you want to solve for y. subtract x/5 from both sides of the equation to get: 5y = m/n - x/5 divide both sides of the equation by 5 to get: y = (m/n)/5 - (x/5)/5 (m/n)/5 is equivalent to m/(5n) (x/5)/5 is equivalent to x/25) your equation becomes: y = m/(5n) - x/25 you can put everything under a common denominator to get: y = 25m/(25*5*n) - 5nx/(25*5*h) which becomes: y = (25m - 5nx) / (125n) you can factor this further to get: y = 5*(5m-nx) / (125n) you can simplify this further to get: y = (5m-nx) / (25n) i think this is as far as you can go. to confirm you did the process correctly, give some numbers to the variables and see if the equations are the same. i let: m = 2 n = 3 x = 4 using these values, the original equation of: x/5 + 5y = m/n becomes: 4/5 + 5y = 2/3 subtract 4/5 from both sides to get: 5y = 2/3 - 4/5 divide both sides by 5 to get: y = (2/3)/5 - (4/5)/5 which becomes: y = 2/15 - 4/25 i'll stop there and see if i can duplicate that with the final equation. the final equation is: y = (5m-nx) / (25n) after substitution this becomes: y = (10 - 12) / 75 i have 2 equations. the final equations shows up as y = (10 - 12) / 75 the original equation shows up as y = 2/15 - 4/25 i will get the original equation to have a common denominator. the original equation becomes: y = 10/75 - 12/75 which becomes (10 - 12) / 75 both equations are identical so the transformation was performed correctly.
 Equations/467081: 1. how do you know when an equation has an infinite number of solutions - show an example. 2. how do you know when an equation has no solution - show an example.1 solutions Answer 320374 by Theo(3464)   on 2011-07-04 16:55:45 (Show Source): You can put this solution on YOUR website!if the equation winds up with an equality and no variables, then you are dealing with an infinite number of solutions. example: 3 = 3 0 = 0 etc. if the equation winds up with no equality and no variables, then you are dealing with no solutions. example: 2 = 3 0 = 5 etc. an example of a system of equations with infinite number of solutions. x + y = 2 2x + 2y = 4 you solve this system of equations by multiplying the first equation by 2 to get: 2x + 2y = 4 (first equation multiplied by 2) 2x + 2y = 4 (second equation) when you subtract the first equation from the second equation, you get: 0 + 0 = 0 which becomes 0 = 0 this indicates an infinite number of solutions. any value for x and any value for y that satisfies one of the equations will automatically satisfy the other equation. for example: if x = 5 and y = -3, then x + y = 2 becomes 5 - 3 = 2 which becomes 2 = 2 which is good. plugging those same values into the second equation gets: 2x + 2y = 4 becomes 2*5 - 2*3 = 4 which becomes 10 - 6 = 4 which becomes 4 = 4 which is good. any combination of x and y that satisfies one of the equation will satisfy the other. an example of no solutions is as follows: x + y = 2 2x + 2y = 7 when you multiply the first equation by 2 to eliminate one of the variables, you wind up eliminating all of the variables and you get: 2x + 2y = 4 (first equation multiplied by 2) 2x + 2y = 7 (second equation) when you subtract the first equation from the second equation, you get: 0 + 0 = 3 which becomes 0 = 3. this is false, so there is no solution to this system of equations. we can graph both the infinite number of solutions and the no solution to show you how the graph will look. your first 2 equations were: x + y = 2 2x + 2y = 4 solve for y in both equations and you will get: y = -x + 2 y = -x + 2 these equations are identical and so their graphs will coincide and look like the same line. your second 2 equations were: x + y = 2 2x + 2y = 7 solve for y in both equations and you will get: y = -x + 2 y = -x + 7/2 these equations have the same slope but have a different y intercept so they are parallel to each other. this means they will never intersect which means you have no common solution. note that all equations are in the slope intercept form. that form is y = mx + b m is the slope and b is the y intercept. if the slopes are the same and the y intercepts are different then the lines are parallel and will never intersect. if the slopes are the same and the y intercepts are the same, then the lines are identical and you have an infinite number of solutions. the graph of the first 2 equations where we had an infinite number of solutions is shown below: the graph of the second 2 equations where we had no solution is shown below: in the first graph, the 2 lines are superimposed on each other because the equations are identical so it looks like you have one line, but you really have 2. the only way to know that is to remove one of the equations from the graph and then you will see that the graph is still there.
 Exponents/466785: 3x^4 ------ y all of this is in parentheses to the -3 power.1 solutions Answer 320077 by Theo(3464)   on 2011-07-03 15:16:27 (Show Source): You can put this solution on YOUR website!there are 3 basic rules of exponentiation used to solve this. the first one is: the second one is: the third one is: you can apply the first rule first and then apply the second rule, or you can apply the second rule first and then apply the first rule. your expression is: applying the first rule first, your expression becomes: applying the second rule next, your expression becomes: applying the third rule next, your expression becomes: we'll go back and do it again and apply the second rule first your expression is: applying the second rule first, your expression becomes: applying the first rule next, your expression becomes: simplifying this, you get: you did not have to apply the third rule in this case.
 Exponents/466787: 4a^-5b^5 -------- 12a^3b^-5 all to the 0 power.. would this just be 1? I know it would simplify to 3b^10 but i dont know what to do with the 0 power ----- a^81 solutions Answer 320068 by Theo(3464)   on 2011-07-03 14:44:45 (Show Source): You can put this solution on YOUR website!it looks to me like your equation would be: anything to the 0 power is equal to 1, so the whole thing should reduce to be equal to 1. you can go through the exercise of simplifying the expression, but if the whole thing is raised to the 0 power, then the answer has to be 1. i show the reduced formula to be equal to: if you follow the rules of exponentiation through, this becomes: which becomes: which becomes: which becomes: which becomes: 1.
 absolute-value/466780: For x < 2; 1/(|x - 2|) + 1/(|2 - x|) = ? I got both terms multiplied by the denominator/denominator of the other term to get both of them to the same common denominator, mainly: (|2 - x| + |x - 2|)/(|2 - x| * |x - 2|), but don't know how to proceed further. Please help1 solutions Answer 320059 by Theo(3464)   on 2011-07-03 14:26:37 (Show Source): You can put this solution on YOUR website!from what i can see, the absolute value of (x-2) is equal to the absolute value of (2-x) for all values of x. this means your denominator will be the same and your equation of: 1/(|x-2|) + 1/(|2-x|) will become (1+1)/(|x-2|) which becomes 2/(|x-2|). if we set this equation equal to y, we get: y = 2/(|x-2|). i believe that's as far as you can go with this. a graph of that equation will look like this: you will have an asymptote at x = 2, because when x = 2, the denominator in the equation equals 0 which makes the value of y undefined. you can see that in the graph. i also graphed your original equation to confirm that i substituted correctly. that equation is: y = 1/(|x - 2|) + 1/(|2 - x|) that graph looks like this: the graphs are identical, so the translation of the formula must have been correct. if you did not recognize that abs(x-2) = abs(2-x), then what you did was in the right direction. fyi: abs(x) is another way to show |x| and is recognized by the graphing software of algebra.com. you would simply set that equation to y and graph it as shown below: the equation that you derived is: y = (abs(x-2) + abs(2-x)) / (abs(x-2)*abs(2-x)) using the algebra.com formula generator, this equation will look like this: y = the graph will look like this: again, the graph is identical, so the transformation of the original equation was done correctly. note that they indicated x < 2 which means that the graph is valid only for values of x < 2 which means all negative values of x + 0, 1, up to, but not including 2. in interval notation, that would be shown as (-infinity, 2) you would effectively ignore the values of x >= 2 in the graph.
 Evaluation_Word_Problems/466775: using the elimination method solve 3x+4y=5 and 2x+y=1. Please show all your work.1 solutions Answer 320052 by Theo(3464)   on 2011-07-03 14:00:36 (Show Source): You can put this solution on YOUR website!the elimination method applies multiplication factors to each equation, as required, so that when you add the equations to each other, or subtract the equations from each other, one of the variables will disappear and you will be left with one equation in one unknown which you can then solve. once you have solved for one of the variables, you can then use that value to help solve for the other variable. your equations are: 3x + 4y = 5 2x + y = 1 if you multiply the second equation by 4, you will be able to eliminate the y from the equation and you can then solve for x. when you multiply an equation, you have to multiply both sides of the equation in order to preserve the equality. multiplying the second equation by 4 gets you: 3x + 4y = 5 (first equation) 8x + 4y = 4 (second equation multiplied by 4) if you subtract the first equation from the second equation, you will be left with: 5x = -1 divide both sides of this equation by 5 and you get: x = -1/5 you can now substitute for x in either of the 2 original equations in order to solve for y. using the first equation of: 3x + 4y = 5 substitute (-1/5) for x to get: 3*(-1/5) + 4y = 5 simplify to get: (-3/5) + 4y = 5 add (-3/5) to both sides of the equation to get: 4y = 5 + (3/5) which becomes: 4y = 28/5 divide both sides of this equation by 4 to get: y = 7/5 the 2 values for x and y are: x = -1/5 y = 7/5 substitute for x and y in the first original equation of: 3x + 4y = 5 to get: 3*(-1/5) + 4*(7/5) = 5 which becomes: -3/5 + 28/5 = 5 which becomes: 25/5 = 5 which becomes: 5 = 5, confirming the values for x and y are solutions for the first equation. substitute for x and y in the second original equation of: 2x + y = 1 to get: 2*(-1/5) + (7/5) = 1 which becomes: -2/5 + 7/5 = 1 which becomes: 5/5 = 1 which becomes: 1 = 1, confirming the values for x and y are solutions for the second equation. those are your answers: x = -1/5 y = 7/5
 Travel_Word_Problems/466774: If im going 200.00 miles per hour, how many miles will I travel in 1 minute?i1 solutions Answer 320038 by Theo(3464)   on 2011-07-03 13:24:04 (Show Source): You can put this solution on YOUR website!you are traveling at 200 miles per hour. divide that by 60 and you will be traveling 200/60 = 3 and 1/3 miles per minute. multiply that by 60 and you get back to 200 miles per hour. there are 60 minutes in an hour.
 Equations/466766: What type of shape is this in the link? http://1.bp.blogspot.com/-xgH3jxjnQyw/ThAlrtxcZnI/AAAAAAAAAAg/uzLAnVQ8RrM/s320/IMG_20110703_011556.jpg Choices: A. regular concave polygon B. convex polygon C. concave polygon D. not a polygon E. regular convex polygon Thank you1 solutions Answer 320036 by Theo(3464)   on 2011-07-03 13:22:47 (Show Source): You can put this solution on YOUR website!here's a good reference that will help you understand convex and concave polygons. http://www.mathopenref.com/polygonconvex.html the convex polygon will not have an internal angle greater than 180 degrees. the concave polygon will. the figure shown in your jpg file looks very much like a concave polygon. there is no such a thing as a regular concave polygon (see the reference). your figure looks very much like a concave polygon. from wikipedia: A simple polygon that is not convex is called concave[2] or reentrant.[3] A concave polygon will always have an interior angle with a measure that is greater than 180 degrees. It is always possible to cut a concave polygon into a set of convex polygons. A polynomial-time algorithm for finding a decomposition into as few convex polygons as possible is described by Chazelle & Dobkin (1985).[4] end of from wikipedia:
 Subset/466514: When \$ 8500 is invested in a savings account paying simple interest for the year, the interest, i in dollars, can be obtained from the equation i = 8500r , where r is the rate of interest in decimal form. Graph i = 8500r , for r up to and including a rate of 16%. If the rate is 12%, how much interest is earned? 1 solutions Answer 319833 by Theo(3464)   on 2011-07-02 07:56:20 (Show Source): You can put this solution on YOUR website!to graph this, we make y = 8500*x where x is the interest rate. since the graph shows x as an integers from 1 to 20, then we need to modify the equation to show as y = 8500 * .01*x. this gets us the decimal equivalent of the percent. 10% = .1 1% = .01 etc. the graph will look like this: i drew a horizontal line at y = 1020 to show you that when x = .12, y = 1020. 8500 * .12 = 1020 which is the simple interest on 8500 when the rate is equal to .12, or 12%. if you trace a vertical line at x = 12, you will see that it intersect the line of the equation at y = 1020.
 test/466456: Which expression must be added to 3x - 7 to equal 0? (A) 0 (B) 3x + 7 (C) -3x + 7 (D) -3x - 7 1 solutions Answer 319831 by Theo(3464)   on 2011-07-02 07:06:07 (Show Source): You can put this solution on YOUR website!3x - 7 = 0 3x = 7 x = 7/3 that just solves for x but doesn't answer the question. your solutions are: A. adding 0 to 3x - 7 results in 3x - 7 so that isn't a solution. B. adding 3x + 7 to 3x + 7 results in 6x + 14 so that isn't a solution. C. adding -3x + 7 to 3x + 7 results in 14 so that isn't a solution. D. adding -3x - 7 to 3x + 7 results in 0 (bingo !!!!!) your equation becomes (3x+7) + (-3x-7) = 3x + 7 - 3x - 7 which results in 0 after you combine all like terms.
 percentage/466504: Chris has two large dogs that want to go out all the time. He has a big fenced in yard so he decided to buy a "doggie door". He did research online and found a pet supply company that was cheapest. He ordered the door online and was given a 20% discount plus a shipping and handling charge of \$4.95. If Chris paid a grand total for the door of \$43.99, what was the original price of the door? Can you please help me figure this problem out. I am having a hard time figuring this problem out. Thank you.1 solutions Answer 319830 by Theo(3464)   on 2011-07-02 06:59:43 (Show Source): You can put this solution on YOUR website!let x = original price of the door. the sale price of the door is equal to x -.2x = .8x the total cost is equal to 43.99 which is the sum of the sale price of the door plus the shipping and handling. this leads to: 43.99 = .8x + 4.95 subtract 4.95 from both sides of this equation to get: 43.99 - 4.95 = .8x simplify this to get: 39.04 = .8x divide both sides of this equation by .8 to get: 39.04 / .8 = x which leads to: x = 39.04 / .8 = 48.8 the original price of the door is \$48.80 20% off the original price leads to a sale price of 39.04 add shipping and handling of 4.95 to get a total price of 39.04 + 4.95 = \$43.99