Solution
Let's let w = the width of the rectangle. If the length is 10 yards less than 3 times the width, then the length is 3w - 10.
How do we find the area? Length times width, right?
So, we have w(3w - 10) = 77. Distributing, we get 3w2 - 10w = 77.
We need to put it all on one side, so we subtract 77 from both sides.
3w2 - 10w - 77 = 0
Now, we have to factor the left side. To do this, we multiply 3(-77) = -231. What are two factors of -231 that add up to -10? How about -21 and 11? -21(11) = -231 and -21 + 11 = -10.
This may not be how you are taught, but we are now going to split -10w into -21w + 11w and plug that into our equation.
Sooooo, now we have 3w2 - 21w + 11w - 77 = 0
Next, we are going to look at the first two terms, 3w2 - 21w. What is common between the two? 3w. Factoring out 3w, we get 3w(w - 7).
Now, the second two terms, 11w - 77. What is common? 11. Factoring out 11, we get 11(w - 7).
Putting all of that together, we now have 3w(w - 7) + 11(w - 7) = 0.
I can explain why and make it more confusing, so I'll just tell you that, since the w - 7 is the same in both parentheses, we keep that as one set of parentheses and combine what's in front into a second set of parentheses.
So, we have (w - 7)(3w + 11) = 0
Lastly, for the left side to equal 0, then either w - 7 and/or 3w + 11 has to equal 0, right?
We pull each out and set them equal to 0.
w - 7 = 0
w = 7
If we plug in 7, we get (7-7)(3(7) + 11) = (0)(21+11) = 0(32) = 0, which is what we want.
3w + 11 = 0
3w = -11
w = -11/3
If we plug -11/3 into the same equation, we'd get 0, again what we want. However, we can't have a negative length! Therefore, -11/3 is out as an answer.
Our width is 7. Since the length is 3w - 10, we plug in 7 and get 3(7) - 10 = 21 - 10 = 11.
Let's check. Length times width = 7(11) = 77. Check!