Video: Simplifying Expressions Using the Commutative and Associative Properties of Addition

A rectangle’s dimensions are (8𝑥 + 4) length units and (16𝑥 + 6) length units. Express the perimeter in terms of 𝑥 and calculate the perimeter when 𝑥 = 1.

02:22

Video Transcript

A rectangle’s dimensions are eight 𝑥 plus four length units and 16𝑥 plus six length units. Express the perimeter in terms of 𝑥 and calculate the perimeter when 𝑥 equals one.

So what I’ve done first is drawn a sketch to help us visualize what’s going on. So we’ve got a rectangle, and we’ve got length 16𝑥 plus six. And we’ve also got width here of eight 𝑥 plus four. And because it’s a rectangle, we know that the opposite sides are equal. So we can add these other values in. So now we’ve got our four sides all labeled. So what do we do next?

Well, what we’re asked to do is to express the perimeter. So let’s think how do we find the perimeter of our rectangle. Well, the perimeter of any shape is the distance round the outside. And we have a formula for the perimeter of a rectangle. And that is 𝑃, our perimeter, is equal to two multiplied by the length plus the width. Or we can also think about it as the perimeter is equal to two lengths plus two widths.

So the perimeter of our shape is going to be equal to two multiplied by. Then we’ve got 16𝑥 plus six plus eight 𝑥 plus four. So we’re gonna get two lots of. And then, first of all, we’ve got 16𝑥 plus eight 𝑥, which is 24𝑥. And then we have positive six and positive four. So it’s gonna be 24𝑥 plus 10. So great, well, have we finished there?

Well, no, what we can do now to simplify even further is to distribute across our parentheses. And when we do that, what we get is 48𝑥 plus 20. And that’s going to be length units as per the question. Okay, great, so we’ve now found an expression for our perimeter. So what we need to do is calculate the perimeter next when 𝑥 is equal to one.

So to find out what the perimeter is when 𝑥 is equal to one, we’re gonna substitute 𝑥 equal to one into our expression. So when we do that, we get the perimeter is equal to 48 multiplied by one plus 20, which gonna give us 68 length units. So therefore, we’ve solved both parts because we’ve expressed our perimeter in terms of 𝑥. And we’ve calculated the perimeter when 𝑥 equals one.

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