The length of a rectangle is 5 inches more than 4 times the width. If the area of the rectangle is 51 square inches, find the length and the width.

Answer 1

You get two equations out of the question:

(Eq 1) L = 4 * W + 5

(Eq 2) L * W = 51

Since the first equation clearly defines L as 4*W+5, let's substitute that for the W in the second equation like this:

(Eq 3): (4 * W + 5)*W = 51

The reason we do this is to reduce on of the two equations to a single variable equation and then solve it. It becomes:

4 * W^2 + 5 * W = 51

or 4 * W^2 + 5 * W - 51 = 0 which is a quadratic equation with coefficients of a=4, b=5, c=-51 and the solutions are W=3 and W=-4.25

I would propose the negative lengths don't make sense for the sides of a rectangle, so let's use W=3 and plug it back into the first equation to find L.

L = 4 * 3 + 5 = 17

Your answer is L = 17 and W = 3, (and the product of the two is 51 as it should be)

[Note: W=-4.25 does work mathematically, producing an L=-12 and an area of 51]

Simplify:

Answer 2

Let x = width. Then length = 4x + 5

x(4x+5) = 51

Solve for x.