Video: Writing and Solving a System of Linear Equations in Three Unknowns

Sophia, Hannah, and Natalie weigh 370 lb in total. If Natalie weighs 20 lb more than Hannah, and Sophia weighs 1.5 times as much as Hannah, how much does each girl weigh?

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Video Transcript

Sophia, Hannah, and Natalie weigh 370 pounds in total. If Natalie weighs 20 pounds more than Hannah and Sophia weighs 1.5 times as much as Hannah, how much does each girl weigh?

So what we’re gonna do is we’re gonna form a series of equations using the information we’ve been given. So, we can say that Sophia weighs 𝑥 pounds, Hannah weighs 𝑦 pounds, and Natalie 𝑧 pounds. So, these three variables are gonna be the ones that we’re gonna be using to form our equations.

So, what’s our first equation going to be? Well, if we take the first bit of information we’ve been given. We’re told that their combined weight is 370 pounds. So therefore, the first equation we can set up is 𝑥 plus 𝑦 plus 𝑧 is equal to 370. And to help us keep track of what we’re going to do, I’m gonna label this equation. So, this will be equation one. Well, what’s the next bit of information we have? The next bit of information we have is that Natalie weighs 20 pounds more than Hannah. So, therefore, we’ve got 𝑧 is equal to 𝑦 plus 20 because it’s 20 more than 𝑦 and 𝑦 is Hannah’s weight. So, therefore, we can say that Natalie weighs 20 pounds more than Hannah. So great. That’s our equation two.

So, what’s our next equation? Well, for our next equation, we need to look at the next bit of information. Now, our next bit of information is that Sophia weighs 1.5 times as much as Hannah. So, therefore, we can say that 𝑥, which is Sophia’s weight, is equal to 1.5𝑦. That’s because we know it is 1.5 times Hannah’s weight. And Hannah’s weight is represented by 𝑦. Okay, great. So, now, we have three equations. We can now use these to find our variables and therefore the weights of the girls.

Well, the first thing we’re going to do is we’re gonna substitute both equation two and equation three into equation one. Because we’ve got a value for 𝑧 in terms of 𝑦 and we have got a value for 𝑥 in terms of 𝑦. And when we do that, what we get is 1.5𝑦 plus 𝑦 plus 𝑦 plus 20 equals 370. So, now, what I can do is simplify the left-hand side. And when we do that, we’re gonna get 3.5𝑦 plus 20 equals 370. So now, if we want to find 𝑦, the first thing we’re gonna do is subtract 20 from each side of our equation. So, when we do that, we’re gonna get 3.5𝑦 equals 350. Then, we divide through by 3.5. So, we get a final value for 𝑦 and that’s 100.

So, now, what we’re gonna do is we’re gonna substitute the value we have for 𝑦 into equations two and three because that’s gonna give us 𝑥 and 𝑧. So, first of all, I’ve substituted into equation two to find 𝑧. So, we’re gonna have 𝑧 is equal to 100 plus 20. So, therefore, we’re gonna get 𝑧 is equal to 120. Then, next, I’m gonna substitute into equation three. And when I do that, I get 𝑥 is equal to 1.5 multiplied by 100. And this is gonna give us an 𝑥-value of 150. Okay, great. So, we found 𝑥, 𝑦, and 𝑧.

So, therefore, we’ve solved the problem because we found the weight of the three girls. So, we’ve got Sophia is equal to 150 pounds, Hannah weighs 100 pounds, and Natalie weighs 120 pounds. And what we can do is run a quick check to make sure it makes sense. And to do that, what we’re gonna do is add together all three girls’ weights. So, we’ve got 150 add 100 add 120. Well, this gives us 370 pounds. So, therefore, we can say yes, we’re definitely correct. And the weights of Sophia, Hannah, and Natalie are 150, 100, and 120 pounds, respectively.

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