The perimeter of a rectangle is 42 feet. the length decreased by three times the width is 1 foot. what are the dimensions of the rectangle?

First you want to make an equation to express the problem. For this one in particular, let's make the length be represented by "L" and the width "W."

The length minus three times the width is 1 foot. This can be represented as L - 3W = 1. This means that L = 3W + 1. This will be our substitute for the length.

The perimeter of a rectangle is 2 times the length plus the width P = 2(L + W). By substituting these variables from what we now know we can find out the dimensions of the rectangle.

42 = 2(3W + 1 + W)
42 = 6W + 2 + 2W     Multiply everything in parenthesis by 2.
42 = 8W + 2              Add the like terms.
40 = 8W                    Subtract 2 on both sides to get the variable (W) by itself.
5 = W                        Divide 8 from 40 to get what W equals.

The width is equal to 5. We can now plug in this answer to the original equation (P = 2(L + W)) or the length equation we made earlier (L = 3W + 1) to find the length.

42 = 2(L + 5)
42 = 2L + 10     Multiply everything in parenthesis by 2.
32 = 2L             Subtract 10 from both sides to get the variable (L) by itself.
16 = L               Divide 2 from 32 to get what L equals.
Or
L = 3(5) + 1      Replace the variable (W) with 5.
L = 15 + 1        Multiply 3 by 5 to get 15.
L = 16              Add the remaining terms!

So the dimensions of the rectangle are 5 and 16, where 5 is the width and 16 is the length. I hope this answer made sense and that you can solve these on your own in the future!