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

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 $\$10050 \cdot 100 = 5000\leftarrow$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? • After you replace$A$with$85-B$you get the equation that they want from you. As a rule, try to use one variable if you can. Mar 11, 2018 at 21:28 ## 2 Answers 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 $$ax+b(n-x)=c.$$

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}