Let the length be [tex]\ell[/tex] and the width be [tex]w[/tex]. 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
[tex]\ell = 2w - 15 cm.[/tex]
Let's plug that into the equation for the perimeter now. The perimeter of a rectangle is [tex]2 \ell + 2w[/tex], so the equation for the perimeter of this rectangle is
[tex]2 \ell + 2w = 102 cm.[/tex]
Plugging in the length in terms of width and solving for the width, we get
[tex]2(2w - 15 cm.) + 2w = 102 cm.[/tex]
[tex]6w - 30 cm. = 102 cm.[/tex]
[tex]6w = 132 cm.[/tex]
[tex]\bf w = 22 cm.[/tex]
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:
[tex]\ell = 2w + 15 cm.[/tex]
[tex]\ell = 2(22 cm.) - 15 cm.[/tex]
[tex]\bf \ell = 29 cm.[/tex]
