Beginning Algebra With Applications, Chapter 4, 4.1, Section 4.1, Problem 44

A carpenter is building a wood door frame. The height of the frame is 1 ft less than three times the width. What is the width of the largest door frame that can be constructed from a board 19 ft long?

If we let $h$ and $w$ be the height and width of the door frame respectively, then we get

$h = 3w-1 \qquad$ Equation 1

And

$w+2h = 19 \qquad$ Equation 2

By substituting equation 1 to equation 2, we have


$
\begin{equation}
\begin{aligned}

w+2 (3w-1) =& 19
\\
w+2 (3w) - 2(1) =& 19
\\
w+6w -2 =& 19
\\
7w =& 21
\\
w =& 3

\end{aligned}
\end{equation}
$


Thus, the width of the door's frame is 3 ft.