Let's assume that the following purchases were made:
Vests = xx
Soccer balls = yy
Cones = y+40y+40 (because 40 more cones were purchased than soccer balls)
The total purchased items = 164164
It can be written in equation form as:
x+y +(y+40) = 164x+y+(y+40)=164
or, x+y+y+40 = 164x+y+y+40=164
or, x+2y+40 = 164x+2y+40=164............(1)
Now the total cost of purchase was = $500$500
Therefore, all items purchased with their individual costs should equal $500$500
This can be represented in an equation as follows:
2.5x+9.25y+ 0.75(y+40) = 5002.5x+9.25y+0.75(y+40)=500
or, 2.5x + 9.25y + 0.75y + 30 = 5002.5x+9.25y+0.75y+30=500, where (0.75xx40 = 30)0.75×40=30)
or, 2.5x+10y+30 = 5002.5x+10y+30=500.................(22)
Now, solving equation (1)(1) and (2)(2)
x+2y+40 = 164x+2y+40=164
2.5x+10y+30 = 5002.5x+10y+30=500
To solve this equation, one of the unknown terms has to be cancelled or removed.
To make that happen, let's multiply x+2y+40 = 164x+2y+40=164 by 2.5 and that will help remove the xx factor from the equation by subtraction.
After multiplication by 2.5, x+2y+40 = 164x+2y+40=164 becomes:
2.5x+5y+100 = 4102.5x+5y+100=410............(3)(3)
Now subtracting equation (3)(3) from equation (22)
cancel2.5x+10y+30 = 500
-(cancel2.5x+cancel5y^(5y)+cancel 100^-70 = cancel410^90)
Therefore, the new equation becomes:
5y-70 = 90
or, 5y = 160 or, y = cancel160^32/cancel5^1 = 32
Thus, the number of soccer balls = 32
Number of cones = 32+40 = 72
and, vests = 164 -72-32 = 60
