The Community Relief Fund receives a large donation of $2800. The foundation agrees to spend the money on $20 school bags, $25 sweaters, $5 books. They want to buy 200 items and send them to schools in earthquake-hit areas. They must order as many books as school bags and sweaters combined. how much of each must they buy?

Let x be the number of school bags, y - the number of sweaters and z - the number of books the foundation wants to buy. They must order as many books as school bags and sweaters combined, so

z = x + y

1 school bag costs $20, then x bags cost $20x,

1 sweater costs $25, then y sweaters cost $25y,

1 book costs $5, then z books cost $5z.

The foundation wants to buy 200 items, then

x + y + z = 200

The foundation wants to spend $2,800, so

20x + 25y + 5z = 2,800

We get the system of three equations:

[tex]\left\{\begin{array}{l}z=x+y\\x+y+z=200\\20x+25y+5z=2,800\end{array}\right.[/tex]

Substitute the first equation into the last two equations:

[tex]\left\{\begin{array}{l}x+y+x+y=200\\20x+25y+5{x+y}=2,800\end{array}\right.\Rightarrow \left\{\begin{array}{l}x+y=100\\25x+30y=2,800\end{array}\right.[/tex]

From the first equation

[tex]x=100-y[/tex]

Substitute it into the second equation

[tex]25(100-y)+30y=2,800\\ \\2,500-25y+30y=2,800\\ \\30y-25y=2,800-2,500\\ \\5y=300\\ \\y=60\\ \\x=100-y=100-60=40\\ \\z=x+y=40+60=100[/tex]

They should buy 40 school bags, 60 sweaters and 100 books.