The scouts sold small and large boxes of cookies as a fund raiser. one scout sold 7 small boxes and 12 large boxes for $54.00. another scout sold 5 small boxes and 10 large boxes for $60.00. the scout leader wrote a system of equations to represent the their sales.

The complete question is
The scouts sold small and large boxes of cookies as a fund raiser. One scout sold 7 small boxes and 12 large boxes for $54.00. Another scout sold 5 small boxes and 10 large boxes for $60.00. The scout leader wrote a system of equations to represent the their sales.
Which constants can each equation be multiplied by so that one variable is eliminated when the equations are added?

let
x> cost small boxes
y> cost large boxes

we know that

7x+12y=54> equation 1
5x+10y=60> equation 2

step 1
multiply equation 1 by (-5/7)
so
(-5/7)*{7x+12y=54}> -5x-(60/7)y=-270/7

-5x-(60/7)y=-270/7
 5x+10y=60

by adding the equations the variable x is eliminated

step 2
multiply equation 1 by (-10/12> -5/6)
so
(-5/6)*{7x+12y=54}>(-35/6)x-10y=-45

(-35/6)x-10y=-45
 5x+10y=60

by adding the equations the variable y is eliminated

step 3
multiply equation 2 by (-7/5)
so
(-7/5)*{5x+10y=60}> -7x-(70/5)y=-84

-7x-(70/5)y=-84
7x+12y=54

by adding the equations the variable x is eliminated

step 4
multiply equation 2 by (-12/10> -6/5)
so
(-6/5)*{5x+10y=60}> -6x-12y=-72

-6x-12y=-72
7x+12y=54

by adding the equations the variable y is eliminated