Question

In the following exercises, solve the problem using a system of equations

The sum of the squares of two numbers is 25. The difference of the numbers is 1. Find the numbers.

Short answer

Two numbers are

$$\begin{gathered} \left(\begin{array}{cc}x=4,\hspace{0.17em}& y=3\\ x=-3,\hspace{0.17em}& y=-4\end{array}\right) \\ 4and3 \\ -3and-4 \end{gathered}$$

Step-by-step solution

Given information 

The sum of the squares of two numbers is 25. The difference of the numbers is 1. Find the numbers.

Frame equations

Lets x be the first number and y be the second number

The sum of the squares of two numbers is 25.

$x^2+y^2=25$

The difference of the numbers is 1.

$x-y=1$

Solve for x

Isolate the second equation for y and substitute it in the first equation.

$$\begin{gathered} x-y=1 \\ x-1=y \\ y=x-1 \end{gathered}$$

Substitute y in the first equation

localid="1667922591167" $$\begin{gathered} x^2+y^2=25 \\ y=x-1 \\ {x}^2+y^2=25 \\ {x}^2+{(x-1)}^2=25 \\ {x}^2+x^2-2x+1=25 \\ 2x^2-2x+1=25 \\ 2x^2-2x-24=0 \\ Divideby2 \\ {x}^2-x-12=0 \\ (x-4)(x+3)=0 \\ x-4=0,x=4 \\ x+3=0 \\ x=-3 \end{gathered}$$

Solve for y 

Substitute the x values and solve for y

$$\begin{gathered} y=x-1 \\ x=4 \\ y=4-1 \\ y=3 \\ x=-3 \\ y=-3-1 \\ y=-4 \end{gathered}$$

solution 

Two numbers are

$$\begin{gathered} 4and3 \\ -3and-4 \end{gathered}$$