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

Dion purchased a new microwave. The diagonal of the door measures 17 inches. The door also has an area of 120 square inches. What are the length and width of the microwave door?

The diagonal of the microwave door measures 17 inches and the area of the door is 120 square inches.

Letlbe the length and w be the width of the microwave door in inches.

The diagonal and the area of the door can be expressed as $\sqrt{l^2+w^2}$and $l\times w$, respectively.

According to the given condition, we have the following equations:

$$\begin{gathered} \sqrt{l^2+w^2}=17 \\ \mathrm{or},l^2+w^2=289--->\left(1\right) \\ l\times w=120--->\left(2\right) \end{gathered}$$

Solving equation (2) for w gives

$w=\frac{120}{l}$

Substituting in (1) gives

$$\begin{gathered} {\left(\frac{120}{w}\right)}^2+w^2=289 \\ \frac{14400}{w^2}+w^2=289 \\ {w}^4-289w^2+14400=0 \\ {w}^4-64w^2-225w^2+14400=0 \\ \left(w^2-64\right)\left(w^2-225\right)=0 \\ \left(w+8\right)\left(w-8\right)\left(w+15\right)\left(w-15\right)=0 \end{gathered}$$

When $w+8=0$, then $w=-8$

When $w-8=0$, then $w=8$

When $w+15=0$, then $w=-15$

When $w-15=0$, then $w=15$

Since width cannot be negative, the acceptable values of w are $w=8,15$

When $w=8$, then from (2), $l=\frac{120}{8}=15$

When $w=15$, then from (2), $l=\frac{120}{15}=8$

Expressing the solution as ordered pair: $\left(15,8\right),\left(8,15\right)$