Question

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

The sum of two numbers is $-45$. Their difference is $-89$. Find the numbers.

Short answer

The required numbers are :-

$-45$and$22$.

Step-by-step solution

Step 1. Given Information

We have given the following problem :-

The sum of two numbers is $-45$. Their difference is $-89$. Find the numbers.

We have to find two number, with sum of numbers is $-45$and difference is $-89$.

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

Step 2. Translate to a system of equations

Let the numbers are :-

$x$and $y$.

We have given that sum of the numbers is $-45$. That is :-

$x+y=-45$.

Also we have difference of the numbers is $-89$. That is :-

$x-y=-89$.

Then we have the following system of equations :-

$\left\{\begin{array}{l}x+y=-45\\ x-y=-89\end{array}\right.$

Step 3. Eliminate one variable 

We have the following system of equations :-

$\left\{\begin{array}{l}x+y=-45\\ x-y=-89\end{array}\right.$

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

$$\begin{gathered} 2x=-45-89 \\ \Rightarrow 2x=-134 \end{gathered}$$

Simplify this equation :-

$$\begin{gathered} 2x=-134 \\ \Rightarrow x=\frac{-134}{2} \\ \Rightarrow x=-67 \end{gathered}$$

So one number is$-67$

Step 4. To find another number   

Now put $x=-67$, in the equation $x+y=-45$, then we have :-

$-67+y=-45$.

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

$$\begin{gathered} -67+y=-45 \\ \Rightarrow y=-45+67 \\ \Rightarrow y=22 \end{gathered}$$

So the second number is$22$.

Step 5. Check the answer   

We find the numbers are :-

$-67$and $22$.

Now add the numbers :-

$-67+22=-45$

This is same as given sum.

Now subtract the numbers :-

$-67-22=-89$.

This is also correct.

So our result is correct.

So the required numbers are :-

$-45and22$.