document.write( "Question 1100019: A sports store sells a total of 70 soccer balls in one month, and collects a total of $2,400. A limited Edition soccer ball is $65 and a Pro NSL soccer ball is $15.Write and solve a system of equations to determine how many of each type of soccer balls were sold. \n" ); document.write( "
Algebra.Com's Answer #714505 by richwmiller(17219)\"\" \"About 
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Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition

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\n" ); document.write( " Lets start with the given system of linear equations
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\n" ); document.write( " \"1%2Ax%2B1%2Ay=70\"
\n" ); document.write( " \"65%2Ax%2B15%2Ay=2400\"
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\n" ); document.write( " In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for y, we would have to eliminate x (or vice versa).
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\n" ); document.write( " So lets eliminate x. In order to do that, we need to have both x coefficients that are equal but have opposite signs (for instance 2 and -2 are equal but have opposite signs). This way they will add to zero.
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\n" ); document.write( " So to make the x coefficients equal but opposite, we need to multiply both x coefficients by some number to get them to an equal number. So if we wanted to get 1 and 65 to some equal number, we could try to get them to the LCM.
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\n" ); document.write( " Since the LCM of 1 and 65 is 65, we need to multiply both sides of the top equation by 65 and multiply both sides of the bottom equation by -1 like this:
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\n" ); document.write( " \"65%2A%281%2Ax%2B1%2Ay%29=%2870%29%2A65\" Multiply the top equation (both sides) by 65
\n" ); document.write( " \"-1%2A%2865%2Ax%2B15%2Ay%29=%282400%29%2A-1\" Multiply the bottom equation (both sides) by -1
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\n" ); document.write( " So after multiplying we get this:
\n" ); document.write( " \"65%2Ax%2B65%2Ay=4550\"
\n" ); document.write( " \"-65%2Ax-15%2Ay=-2400\"
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\n" ); document.write( " Notice how 65 and -65 add to zero (ie \"65%2B-65=0\")
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\n" ); document.write( " Now add the equations together. In order to add 2 equations, group like terms and combine them
\n" ); document.write( " \"%2865%2Ax-65%2Ax%29%2B%2865%2Ay-15%2Ay%29=4550-2400\"
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\n" ); document.write( " \"%2865-65%29%2Ax%2B%2865-15%29y=4550-2400\"
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\n" ); document.write( " \"cross%2865%2B-65%29%2Ax%2B%2865-15%29%2Ay=4550-2400\" Notice the x coefficients add to zero and cancel out. This means we've eliminated x altogether.
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\n" ); document.write( " So after adding and canceling out the x terms we're left with:
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\n" ); document.write( " \"50%2Ay=2150\"
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\n" ); document.write( " \"y=2150%2F50\" Divide both sides by \"50\" to solve for y
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\n" ); document.write( " \"y=43\" Reduce
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\n" ); document.write( " Now plug this answer into the top equation \"1%2Ax%2B1%2Ay=70\" to solve for x
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\n" ); document.write( " \"1%2Ax%2B1%2843%29=70\" Plug in \"y=43\"
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\n" ); document.write( " \"1%2Ax%2B43=70\" Multiply
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\n" ); document.write( " \"1%2Ax=70-43\" Subtract \"43\" from both sides
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\n" ); document.write( " \"1%2Ax=27\" Combine the terms on the right side
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\n" ); document.write( " \"cross%28%281%2F1%29%281%29%29%2Ax=%2827%29%281%2F1%29\" Multiply both sides by \"1%2F1\". This will cancel out \"1\" on the left side.
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\n" ); document.write( " \"x=27\" Multiply the terms on the right side
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\n" ); document.write( " So our answer is
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\n" ); document.write( " \"x=27\", \"y=43\"
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\n" ); document.write( " which also looks like
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\n" ); document.write( " (\"27\", \"43\")
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\n" ); document.write( " Notice if we graph the equations (if you need help with graphing, check out this solver)
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\n" ); document.write( " \"1%2Ax%2B1%2Ay=70\"
\n" ); document.write( " \"65%2Ax%2B15%2Ay=2400\"
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\n" ); document.write( " we get
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\n" ); document.write( " graph of \"1%2Ax%2B1%2Ay=70\" (red) \"65%2Ax%2B15%2Ay=2400\" (green) (hint: you may have to solve for y to graph these) and the intersection of the lines (blue circle).
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\n" ); document.write( " and we can see that the two equations intersect at (\"27\",\"43\"). This verifies our answer.

\n" ); document.write( "\n" ); document.write( "x=27 at $65, y=43 at $15 \n" ); document.write( "
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