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We have given the following problem :-

The sum of two numbers is $37$. Their difference is $9$. Find the numbers.

We have to find two number, with sum of numbers is $37$and difference is $9$.

Firstly we will translate this problem to a system of equation. After that we will apply elimination method to solve the problem.

Let the numbers are :-

$x$and $y$.

We have given that sum of the numbers is $37$. That is :-

$x+y=37$.

Also we have difference of the numbers is $9$. That is :-

$x-y=9$.

Then we have the following system of equations :-

$\left\{\begin{array}{l}x+y=37\\ x-y=9\end{array}\right.$

We have the following system of equations :-

$\left\{\begin{array}{l}x+y=37\\ x-y=9\end{array}\right.$

Now to eliminate one variable $y$, add both the equations, then we have :-

$2x=46$.

Simplify this equation :-

$$\begin{gathered} 2x=46 \\ ⇒x=\frac{46}{2} \\ ⇒x=23 \end{gathered}$$

So one number is $23$.

Now put $x=23$, in the equation $x+y=37$, then we have :-

role="math" $23+y=37$.

Now solve the equation for $y$, then we have :-

$$\begin{gathered} 23+y=37 \\ ⇒y=37-23 \\ ⇒y=14 \end{gathered}$$

So the second number is$14$.

We find the numbers are :-

$23$and$14$.

Now add the numbers :-

$23+14=37$

This is same as given sum.

Now subtract the numbers :-

$23-14=9$.

This is also correct.

So our result is correct.

So the required numbers are :-

$23and14$.