How can I solve the statement, without a system of equations?

Question

I have this statement:

Steve saving money, in total he saved $85$ coins, divided between $\$100$ and $\$500$ coins.

Also together a figure of $\$22500$ in total.

So, how much money did you put together, in $\$100$ coins?

With system of equations:

$A + B = 85$

$100A + 500B = 22500$

$A = 85 - B$

$100(85 - B) + 500B = 22500$

$8500 - 100B + 500B = 22500$

$400B = 14000$

$B = 35$

so, $A = 85 - 35 = 50$ , $50$ coins of $\$100$

$50 \cdot 100 = 5000\leftarrow$ This result is good.

But they asked me to do it with another form, which is an equation:

If $n$ objects, that have a value of $c$, that are composed of $x$ objects, which each one have a value of $a$, and $n - x$ objets that have a value of $b$, the equation to find $x$ is:

$ax + b(n-x) = c$

I really did not understand it at all. Could you explain to me how it works? And what is it about?

Answer 1

Both approaches are exactly the same. In your problem, $x=B$, $n=85$, $a=500$, $b=100$, and $c=22500$. Also, you yourself figured out that $A= 85-B=n-x$. Consequently, $$100A+500B = 22500$$ is equivalent to $$100(85-B)+500B=22500,$$ which is $$ax+b(n-x)=c.$$

Answer 2

Perhaps that they want you to solve it this way:\begin{align}ax+b(n-x)=c&\iff(a-b)x=c-bn\\&\iff x=\frac{c-bn}{a-b}.\end{align}