Question

The length of a rectangle is 2 inches less than 4 times the width. The perimeter is 86 inches. Find the length and the width.The length is ? inches and the width is ? inches.

Answer 1

The perimeter of a rectangle is given by

`p=2l+2w`

Where

p - perimeter

l - length

w - width

We know that the length is 2 inches less than 4 times the width, then

`l=4w-2`

We also know that the perimeter is 86 inches, then

`86=2l+2w`

Now we have two equations:

`\begin{cases}l=4w-2 \\ 86=2l+2w\end{cases}`

Let's just rewrite the equations

`\begin{cases}l-4w=-2 \\ 2l+2w=86\end{cases}`

Now we have a system of equations, let's multiple the first equation by -2

`\begin{cases}-2l+8w=4 \\ 2l+2w=86\end{cases}`

Now we sum the equations

`\begin{gathered} -2l+2l+8w+2w=86+4 \\ \\ 10w=90 \\ \\ w=9\text{ inches} \end{gathered}`

Now we already know that the width is 9 inches. We can find the length using the equation:

`\begin{gathered} l=4w-2 \\ \\ l=4\cdot9-2 \\ \\ l=34\text{ inches} \end{gathered}`

The length is 34 inches and the width is 9 inches.