Let the length be \(\ell\) and the width be \(w\). Now, write the length in terms of the width. "The length is 15 cm. less than twice the width" in the form of an equation is
\(\ell = 2w - 15 cm.\)
Let's plug that into the equation for the perimeter now. The perimeter of a rectangle is \(2 \ell + 2w\), so the equation for the perimeter of this rectangle is
\(2 \ell + 2w = 102 cm.\)
Plugging in the length in terms of width and solving for the width, we get
\(2(2w - 15 cm.) + 2w = 102 cm.\)
\(6w - 30 cm. = 102 cm.\)
\(6w = 132 cm.\)
\(\bf w = 22 cm.\)
To find the length, we just have to plug in the width into the equation we wrote for the length in terms of the width:
\(\ell = 2w + 15 cm.\)
\(\ell = 2(22 cm.) - 15 cm.\)
\(\bf \ell = 29 cm.\)
