SOLUTION: 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

Algebra ->  Coordinate Systems and Linear Equations  -> Linear Equations and Systems Word Problems -> SOLUTION: 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       Log On


   

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.
Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!

Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

1%2Ax%2B1%2Ay=70
65%2Ax%2B15%2Ay=2400

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).

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.

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.

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:

65%2A%281%2Ax%2B1%2Ay%29=%2870%29%2A65 Multiply the top equation (both sides) by 65
-1%2A%2865%2Ax%2B15%2Ay%29=%282400%29%2A-1 Multiply the bottom equation (both sides) by -1


So after multiplying we get this:
65%2Ax%2B65%2Ay=4550
-65%2Ax-15%2Ay=-2400

Notice how 65 and -65 add to zero (ie 65%2B-65=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%2865%2Ax-65%2Ax%29%2B%2865%2Ay-15%2Ay%29=4550-2400

%2865-65%29%2Ax%2B%2865-15%29y=4550-2400

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.



So after adding and canceling out the x terms we're left with:

50%2Ay=2150

y=2150%2F50 Divide both sides by 50 to solve for y



y=43 Reduce


Now plug this answer into the top equation 1%2Ax%2B1%2Ay=70 to solve for x

1%2Ax%2B1%2843%29=70 Plug in y=43


1%2Ax%2B43=70 Multiply



1%2Ax=70-43 Subtract 43 from both sides

1%2Ax=27 Combine the terms on the right side

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.


x=27 Multiply the terms on the right side


So our answer is

x=27, y=43

which also looks like

(27, 43)

Notice if we graph the equations (if you need help with graphing, check out this solver)

1%2Ax%2B1%2Ay=70
65%2Ax%2B15%2Ay=2400

we get



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).


and we can see that the two equations intersect at (27,43). This verifies our answer.

x=27 at $65, y=43 at $15