When looking at these problems, making a picture is a great first step. Let W = the rectangle's width and let L = the rectangle's width and let's translate "the length of a rectangle is 7 yd less than three times the width" into words.
The length ---- L
is ---- =
7 less than three times -------- 3L -7
It's tempting to say 7 - 3L for the other side, but the 7 is subtracted after the multiplying, so we "write it backwards".
_______L_______
| |
| |
| | 3L - 7
| |
| |
_______________
We used L above to that we have an equation with only variable.
Since our area is 66 square yards, then we create our equation to solve.
66 = L * (3L-7)
66 = 3L²- 7L
3L² - 7L - 66 = 0
We moved all the variables to one and set it equal to zero since it's a second degree equation. Now we can use factoring/quadratic formula and the Zero Product Property. The equation does factor but if it's not obvious, go to the quadratic formula with a = 3, b = -7, and c = -66. (and consider what's in asterisks as TL DR)
*****
We are looking for a pair of numbers that multiply to 66, and our pairs are 1 & 66, 2 & 33, 3 & 22, and 6 & 11. We are also looking for a pair that multiply to 3, and only 3 & 1 work. Since there's a negative sign with the 66, the signs will be opposite - one plus and one minus. This gives this information.
(3L + ___) (L - ____) or (3L - ____) (L + _____)
So we have need from the pairs that subract to seven (the middle term) when one is tripled. 6 and 33 (triple one) won't work but 18 (triple the other one) and 8 will work. If you started left to right, 1 and 198 or 3 and 66 would be your first pairs.
(3L + 11) (L - 6) are our factors. We check it through FOIL to make sure it works.
3x² - 18x + 11x - 66 = 3x² - 7x - 66
So
3L² - 7L - 66 = 0
(3L + 11) (L - 6) = 0 by factoring
3L + 11 = 0 or L - 6 = 0 by the Zero Product Property
3L = -11 or L = 6
L = -11/3 or L = 6.
*****
L = -11/3 might work as a solution but it is not realistic in this problem. Rectangles have positive lengths, so the only solution we keep is L = 6.
Back to the picture. If we know L = 6, then to find the width we back substitute it in for the width. 3 * 6 - 7 = 18 - 7 = 11.
Therefore, the rectangle is 6 yards long by 11 yards wide.
