Video: Forming and Solving a System of Linear Equations with Three Unknowns

Three numbers sum up to 147. The smallest number is half the middle number, which is half the largest number. What are the three numbers?

03:14

Video Transcript

Three numbers sum up to 147. The smallest number is half the middle number, which is half the largest number. What are the three numbers?

To help us solve this problem, I’m gonna start by denoting our smallest number as 𝑥. Then if we look back at the question, we can see that the smallest number is actually half the middle number. So therefore, we can denote the middle number as two 𝑥, because actually another way of saying that the smallest number is half the middle number is to saying the middle number is actually two times as big as the smallest number, so in that case two 𝑥 cause it’s twice as big as 𝑥.

And then finally, we’re told that, actually, the middle number is half the largest number. So we can say that the largest number is actually equal to four 𝑥. And it’s actually four 𝑥 because two 𝑥 multiplied by two is equal to four 𝑥. And it says that the middle number is half the largest number. So therefore, if we multiply the middle number by two, that will give us our largest number.

Okay, great! So we actually now have terms for each of our numbers. If we look at the start of our question, it actually tells us that the three numbers sum up to 147. So therefore, we can actually form an equation. We can say that 𝑥 plus two 𝑥 plus four 𝑥 is equal to 147. And this is because it says the smallest number plus the middle number plus the largest number is equal to 147. So then we’re gonna collect the terms, which is gonna give us that seven 𝑥 is equal to 147.

Okay, great! We can now solve this by dividing both sides by seven. And this gives us an answer of 𝑥 is equal to 21. So, great! Have we solved the problem? Is this it? Is this all we need to do? Well, no, because if we actually look back at our question, it says, “What are the three numbers?”

Be very careful with this. This is where mistakes are often made because people actually think, “Right, great! I’ve found our 𝑥; that’s it; that’s me done.” But no, we actually need to find out what each of the three numbers actually are.

Our smallest number, which we denoted by 𝑥, is equal to 21. Our middle number, which we denoted by two 𝑥, is equal to two multiplied by 21, because it’s two 𝑥, so two multiplied by 𝑥, which is going to be 42. And finally, our largest number, which we denoted by four 𝑥, is gonna be equal to four multiplied by 21 because it was four multiplied by 𝑥, which gives us a value of 84.

So therefore, we can say the three numbers are 21, 42, and 84. And just to double-check our answer, we can double-check it by looking at the sum, because we know that the sum is meant to be 147. So if we have 21 plus 42 plus 84, we get 147. So yes, great! All checked. We’re happy with our answers that say that the smallest number is 21, the middle number is 42, and the largest number is 84.

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