Video: Writing and Solving Linear Equations in a Geometric Context

Video Transcript

Find the length of 𝑋𝑌.

Here would be the distance of the segment 𝑋𝑌 which is equal to six 𝑎 centimeters. So that means we need to solve for 𝑎. That way, we can take 𝑎 multiplied by six. And we will know the length of 𝑋𝑌. So in order to solve for 𝑎, we need to somehow create an equation. The distance from 𝑋 to 𝑌 plus the distance from 𝑌 to 𝑍, which is 12 centimeters, is equal to the entire distance from 𝑋 to 𝑍. And 𝑋𝑍 can be represented by eight 𝑎 minus two centimeters.

This would be like taking a piece of string and cutting it into two pieces. And now that our string is cut into two pieces, it can be represented by our blue and green segments, 𝑋𝑌 and 𝑌𝑍. And the original string, before we cut it, would be represented by 𝑋𝑍, the entire thing. So when we cut the string, the original distance didn’t change. We just cut it into two little pieces. So that’s what’s being represented here. So let’s begin plugging in for these segments. That way we can solve for 𝑎.

𝑋𝑌 can be replaced with six 𝑎 centimeters. 𝑌𝑍 can be replaced with 12 centimeters. And 𝑋𝑍 can be replaced with eight 𝑎 minus two centimeters. So since we’re going to be solving for 𝑎, let’s leave out the measurements. And since it’s a length, our length should be in centimeters, leaving us with an equation of six 𝑎 plus 12 equals eight 𝑎 minus two.

So we need to solve for 𝑎. Let’s go ahead and subtract both sides of the equation by six 𝑎. And the reason for moving the 𝑎s to the right-hand side is if we move it to the right-hand side, 𝑎 will stay positive. If we would subtract eight 𝑎 to the left-hand side, 𝑎 would become negative, which would be okay. But if our variable turns negative, sometimes there might be an extra step of getting rid of the negative sign. But it would be perfectly fine.

So moving on, the six 𝑎s cancel. So we have 12 equals, eight 𝑎 minus six 𝑎 is two 𝑎, and then minus two. So to solve for 𝑎, we add two to both sides of the equation. And we get 14 equals two 𝑎. Since we’re running out of room, let’s move up here. So we have that 14 equals two times 𝑎. So we need to isolate 𝑎, get it by itself. So to do that, we need to get rid of two. And two is being multiplied to 𝑎. And the opposite of multiplying would be to divide. And dividing both sides by two, we get that seven is equal to 𝑎.

So now we can use this to find the length of segment 𝑋𝑌. Segment 𝑋𝑌 is equal to six 𝑎 centimeters. So let’s plug in seven for 𝑎. And six times seven is equal to 42. So the length of segment 𝑋𝑌 is equal to 42 centimeters.