Question

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

The sum of two numbers is $65$. Their difference is $25$. Find the numbers.

Short answer

The required numbers are :-

$45and20$.

Step-by-step solution

Step 1. Given Information

We have given the following problem :-

The sum of two numbers is 65. Their difference is 25. Find the numbers.

We have to find two number, with sum of numbers is $65$and difference is$25$.

Firstly we will translate this problem to a system of equation. After that we will apply elimination method to solve the problem.$65$

Step 2. Translate to a system of equations

Let the numbers are :-

$x$and $y$.

We have given that sum of the numbers is $65$. That is :-

$x+y=65$.

Also we have difference of the numbers is $25$. That is :-

localid="1647698334988" $x-y=25$

Then we have the following system of equations :-

$\left\{\begin{array}{l}x+y=65\\ x-y=25\end{array}\right.$

Step 3. Eliminate one variable

We have the following system of equations :-

$\left\{\begin{array}{l}x+y=65\\ x-y=25\end{array}\right.$

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

$2x=90$.

Simplify this equation :-

$$\begin{gathered} 2x=90 \\ \Rightarrow x=\frac{90}{2} \\ \Rightarrow x=45 \end{gathered}$$

So one number is$45$.

Step 4. To find another number

Now put $x=45$, in the equation $x+y=65$, then we have :-

role="math" $45+y=65$.

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

role="math" localid="1647699420654" $$\begin{gathered} 45+y=65 \\ \Rightarrow y=65-45 \\ \Rightarrow y=20 \end{gathered}$$

So the second number isrole="math" localid="1647699426146" $20$.

Step 5. Check the answer

We find the numbers are :-

$45$and $20$

Now add the numbers :-

role="math" $45+20=65$

This is same as given sum.

Now subtract the numbers :-

$45-20=25$.

This is also correct.

So our result is correct.

So the required numbers are :-

$45and20$