In the following exercises, translate to a system of equations and solve.

The sum of two numbers is $-27$. Their difference is $-59$. Find the numbers.

We have given the following problem :-

The sum of two numbers is $-27$. Their difference is $-59$. Find the numbers.

We have to find two number, with sum of numbers is $-27$and difference is $-59$.

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 $-27$. That is :-

$x+y=-27$.

Also we have difference of the numbers is $-59$. That is :-

$x-y=-59$.

Then we have the following system of equations :-

$\left\{\begin{array}{l}x+y=-27\\ x-y=-59\end{array}\right.$

We have the following system of equations :-

$\left\{\begin{array}{l}x+y=-27\\ x-y=-59\end{array}\right.$

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

$$\begin{gathered} 2x=-27-59 \\ \Rightarrow 2x=-86 \end{gathered}$$

Simplify this equation :-

$$\begin{gathered} 2x=-86 \\ \Rightarrow x=\frac{-86}{2} \\ \Rightarrow x=-43 \end{gathered}$$

So one number is$-43$.

Now put $x=-43$, in the equation $x+y=-27$, then we have :-

$-43+y=-27$.

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

$$\begin{gathered} -43+y=-27 \\ \Rightarrow y=-27+43 \\ \Rightarrow y=16 \end{gathered}$$

So the second number is $16$.

We find the numbers are :-

$-43$and $16$.

Now add the numbers :-

$-43+16=-27$

This is same as given sum.

Now subtract the numbers :-

$-43-16=-59$.

This is also correct.

So our result is correct.

So the required numbers are :-

$-43and16$.