Question

The sum of two numbers is $7$, and the difference of their

squares is $21$. Find the numbers.

Short answer

The required numbers are,$5,2$.

Step-by-step solution

Step 1. Given information

Given that,

The sum of two numbers is $7$. So, $a+b=7$.

The difference of their squares is $21$. So, $a^2-b^2=21$.

So, the system of equations are,

$\left\{\begin{array}{l}a+b=7\\ a^2-b^2=21\end{array}\right.$

Step 2 Let us write the first equation as,

$a=7-b$.

Substitute the value of $a$in the second equation.

$$\begin{gathered} {\left(7-b\right)}^2-b^2=21 \\ {7}^2-14b+b^2-b^2=21 \\ 49-14b=21 \\ 14b=28 \\ b=2 \end{gathered}$$

Step 3 Now substitute the value of b in the first equation.

$$\begin{gathered} a+2=7 \\ a=7-2 \\ a=5 \end{gathered}$$

So, the required numbers are,$5,2$.