Question

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

The difference of the squares of two numbers is 45. The difference of the square of the first number and twice the square of the second number is 9. Find the numbers.

Short answer

Two numbers are

9 and 6

-9 and 6

9 and -6

-9 and -6

Step-by-step solution

Given Information 

The difference of the squares of two numbers is 45. The difference of the square of the first number and twice the square of the second number is 9. Find the numbers.

Set up equations 

Let x and y are the two unknown numbers

Set up equations using the given information

The difference of the squares of two numbers is 45

$x^2-y^2=45$

The difference of the square of the first number and twice the square of the second number is 9

$x^2-2y^2=9$

solve the equation for y 

$$\begin{gathered} Eliminatethevariablex \\ {x}^2-y^2=45 \\ {x}^2-2y^2=9 \\ Multiplythesecondequationby-1 \\ {x}^2-y^2=45 \\ -x^2+2y^2=-9 \\ -------- \\ {y}^2=36 \\ y=\sqrt{36} \\ y=\pm 6 \end{gathered}$$

solve for x

$$\begin{gathered} x^2-y^2=45 \\ y=-6 \\ Substitutethevalueandsolveforx \\ {x}^2-{(-6)}^2=45x^2-36=45 \\ {x}^2=45+36 \\ {x}^2=81 \\ x=\sqrt{81} \\ x=\pm 9 \\ y=6 \\ {x}^2-{\left(6\right)}^2=45 \\ {x}^2-36=45 \\ {x}^2=45+36 \\ {x}^2=81 \\ x=\sqrt{81} \\ x=\pm 9 \end{gathered}$$

solution 

Two numbers are

9 and 6

-9 and 6

9 and -6

-9 and -6