Video: Using Linear Equations to Solve Problems

The lengths (in centimeters) of the sides of a hexagon have consecutive integer values, and the hexagon’s perimeter is 147 cm. How long is the longest side?

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Video Transcript

The length in centimeters of the sides of a hexagon have consecutive integer values, and the hexagon’s perimeter is 147 centimeters. How long is the longest side?

We know that a hexagon has six sides. We do not know the length of any of the six sides. But we do know they add up, or the perimeter, the distance all the way around, is 147 centimeters. We also know that each side has a consecutive integer value. And that means if one of the side lengths is five, the next would be six, seven, and all the way up until the sixth side.

But since we don’t know the smallest side’s length, we’ll write 𝑥 for its length. The second side length would then be 𝑥 plus one; it’s one larger than the smallest side. Side three would be 𝑥 plus one plus one, and we can simplify that to say 𝑥 plus two. Side four is the shortest side plus three, 𝑥 plus three. Side five is the shortest side plus four, 𝑥 plus four. And side six is the shortest side plus five, 𝑥 plus five.

When we add all of the sides together, they’ll equal 147 centimeters. If we solve for 𝑥, we’ll be able to find all of the side lengths. We combine like terms. There are six 𝑥s. We can turn that into six 𝑥, and then we add the whole numbers: one plus two plus three plus four plus five equals 147.

One plus two plus three plus four plus five equals 15. Bring down our six 𝑥. Six 𝑥 plus 15 equals 147. To get 𝑥 by itself, we subtract 15 on both sides. Plus 15 minus 15 equals zero. 147 minus 15 equals 132. Now we can divide both sides of our equation by six. 𝑥 times six divided by six equals 𝑥. 132 divided by six equals 22. Since 𝑥 equals 22, the shortest side of this hexagon is 22 centimeters.

We are primarily interested in how long the longest side is. To find the longest side, we take the shortest side and add five. 22 plus five equals 27. The longest side on this hexagon measures 27 centimeters.

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