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Perimeter of a Rectangle, Two Unknowns

Q: The perimeter of a rectangle is 74 inches. If the width is doubled and the length is halved, the new rectangle will have a perimeter is 76 inches. What are the dimensions?

A: You have two unknowns, l=length and w=width, so you need two equations.

Recall that perimeter of a rectangle is P = 2l + 2w

Eqn 1: 74 = 2w + 2l
Eqn 2: 76 = 2(2w) + 2l/2
76 = 4w +l
Rearrange Eqn. 2, isolating for l:

l=76-4w

Insert this expression for l into Eqn 1:

74 = 2w + 2(76-4w)
74 = 2w + 152 -8w
6w = 78
w=13

The length can be found from either Eqn 1 or 2:
l = 76-4w
l = 76-4(13)
l=24

So the width is 13 inches and the length is 24 inches.