Derek D. answered • 08/19/21
Derek: Algebra/Statistics Tutor
To find the perimeter of a rectangle, we add up the lengths of it's four sides. Rectanges have two pairs of equal sides, two horizontal and two vertical. We can call the length of the horizontal edges the 'width' of the rectangle, and the length of the vertical edges the 'length' of the rectangle.
So to add up all four sides, we would have to 'width' edges and two 'length' edges. We have the following formula that's meant to be a shorthand for this concept:
P = 2W + 2L
P means total Perimeter, W means width, L means Length, and if there's a 2 next to a letter it means we're looking two copies of that side. (Since the 'left' and 'right' sides of any rectangle are the same, then you can think of them as copies).
In Algebra, we often want to do two things with formulas:
- Fill in what we know. In this case, we're going to take the fomula and replace 'P' with '24' because in our prompt, the perimeter was told to us that it is 24.
- Substitute variables. When we say "the length is 4 greater than the width", like the original prompt says, then we can express that mathmatically by saying "L = 4 + width" or "W + 4 = L" (You could write it a couple of different ways, those are two of many.) If L is really equal to '4 + W', we should be able to exchange them for each other. So L can be 'exchanged' or 'substituted' for '4 + W'.
So our formula goes from:
P = 2W + 2L
to
24 = 2W + 2(4 + W)
Notice how the '2W' stays the same, because we aren't making any substitutions. Also notice how I added parentheses when I substituted the 'L' for '4 + W'. Since 2 was being multiplied with L before, now 2 is being multiplied with '4 + W'.
Now that our formula only has one variable, we can solve for it!
24 = 2W + 2(4 + W) Original Formula
24 = 2W + 8 + 2W Distribute
24 = 4W + 8 Combine Like Terms
-8 - 8 Subtract 8 to both sides
16 = 4W Now we divide both sides by 4 to isolate W
4 = W Final Product!
"4 = W" means that the Width of our Rectangle is '4', since W in our original formula meant the Width of our rectangle, and we can now see that it is equal to 4. So what can we do with that info? We can substitute (kinda like an exchange) that W for a 4 in our original equation!
p = 2W + 2L Original Equation
24 = 2(4) + 2L Substitute what we know (P is 24, W is 4)
24 = 8 + 2L Multiply/Simplify the 2(4)
16 = 2L Subtract 8 from both sides
8 = L Divide both sides by 2
"8 = L"! That means the length of our rectangle is 8. Now we have both sides! 8 = L, and 4 = W. Are we done? The problem says to 'Find the dimensions'. In a rectangle, that's length and width, and that's what we have! So our final answer is L = 8 and W = 4.
