11. find the value of x for the rectangle to the right if the perimeter is
(18x – 20) in. use the value for x and find the lengths of the two sides
and the perimeter of the rectangle.
x=
side lengths: and
perimeter:

In order to find the answer to this one, we're going to need to do a lot of algebra. First of all, remember that a rectangle has two sets of congruent sides, which make the perimeter when put together. I'll define the two different measurements (one for one set, one for the other set) as l and w. Therefore, 2l + 2w = P, or the two sets of congruent sides equal the perimeter.

In this problem, l = x + 12 and w = 2x + 8. The perimeter is 18x - 20. Let's plug these values into our earlier equation.

2(x + 12) + 2(2x + 8) = 18x - 20   Substitution 
2x + 24 + 4x + 16 = 18x - 20   Distribute
6x + 40 = 18x - 20   Combine like terms
6x = 18x - 60   Subtract 40 from both sides
-12x = -60   Subtract 18x from both sides
x = 5   Divide both sides by -12

Therefore, x = 5. Let's plug that in to l (x + 12) and w (2x + 8) to find the side lengths. 

5 + 12   Substitution
17   Add

The length of one side is 17 units.

2(5) + 8   Substitution
10 + 8   Multiply
18   Add

The length of the other side is 18 units.

Now, let's plug 5 into the perimeter (18x - 20).

18(5) - 20   Substitution
90 - 20   Multiply
70   Subtract

Therefore, the perimeter is 70 units. We can check this by plugging into the earlier equation we made to find x.

2l + 2w = 70   Given
2(17) + 2(18) = 70   Substitute each side length
34 + 36 = 70   Multiply
70 + 70   Add

Since the equation is true, our math checks out. Our answer is correct!

Therefore, x = 5, the side lengths are 17 and 18 units, and the perimeter is 70 units.

Hope this helps!