Question

A rectangle has a length of 22 inches less than 7 times its width. If the area of the rectangle is 3197 square inches, find the length of the rectangle.

Answer 1

Like any other area problem, we start with the basic formula: length x width = area. This is the basis of what we can substitute and whatnot.

length x width = area
Read through the sentence and look for the words "length", "width" or "area". Chances are, the sentence gives you at least one definite value of the terms. Once you found the values, substitute them into the equation.
length x width = 3197
Now it's time to get a little creative. In algebra, we use variables to represent anything we don't know, or what we're trying to find out. In this case we can either represent the length of width with a variable. However, it would be easier to represent the width with a variable.

width = x
length = (7x - 22)
We have values we can substitute into the equation yet again!
x(7x-22) = 3197
7x^2 - 22x - 3197 = 0
(7x + 139)(x - 23) = 0
(All I did above was factor the quadration equation to find two solutions for x.)
x = 23
Notice how I chose the second binomial instead of the first. This is because the x value would be positive instead of negative, and a negative length is not possible.

Therefore, the width = 23 inches, and the length = 139 inches

Answer 2

The area formula for a rectangle is A = L*W. We have the area as 3197, now we just have to find a way to directly relate the length to the width with only one unknown, since we can't have 2 unknowns. The problem tells us that the length is 22 in. less than 7xW. That will be expressed as L=7W-22, and width is just W. Filling in we have 3197=W(7W-22). Distributing through the parenthesis we have
`3197=7W^2-22W`. Since we have to solve for x and the only way we can do that is by factoring, we will move the 3197 over to the other side by subtraction and then factor the quadratic.
`7W^2-22W-3197=0`. I recommend putting that into the quadratic formula, and when you do you get W values of 23 and -19.857 The only 2 things in math that will never EVER be negative are time and distance/length. And since one of our lengths there is negative we know that that is not the one we will use. Therefore, the width is 23. The length then is L=7(23)-22 which is 139