# The average of two numbers is 50. Their difference is 40, how do you write an equation that may be used to find x the smallest of the two numbers?

Aug 4, 2015

$x = 30$

#### Explanation:

You know that you need to find two numbers, $x$ and let's say $y$.

The average of two numbers is equal to their sum divided by $2$, so your first equation will be

$\frac{x + y}{2} = 50$

The difference between $y$ and $x$, since $x$ is the smallest of the two, is equal to $40$, which means that your second equation will be

$y - x = 40$

You thus have a system of two equations

$\left\{\begin{matrix}\frac{x + y}{2} = 50 \\ y - x = 40\end{matrix}\right.$

To solve for $x$, use the first equation to express $y$ as a function of $x$

$\frac{x + y}{2} = 50 \iff x + y = 100 \implies y = 100 - x$

Plug this into the second equation to get

$\left(100 - x\right) - x = 40$

$\textcolor{b l u e}{100 - 2 x = 40}$ $\to$ this is the equation that will get you $x$.

The value of $x$ will be

$2 x = 60 \implies x = \textcolor{g r e e n}{30}$

The value of $y$ will be

$y = 100 - 30 = \textcolor{g r e e n}{70}$