# Video: Writing and Solving a System of Linear Equations in Three Unknowns in a Real-World Context

William has lemon, orange, and pomegranate trees in his backyard. An orange weighs 8 oz, a lemon 5 oz, and a pomegranate 11 oz. William picked 142 pieces of fruit weighing a total of 70 lb 10 oz. Given that he picked 15.5 times more oranges than pomegranates, how many of each of the three fruits did William pick?

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### Video Transcript

William has lemon, orange, and pomegranate trees in his backyard. An orange weighs eight ounces, a lemon five ounces, and a pomegranate 11 ounces. William picked 142 pieces of fruit, weighing a total of 70 pounds 10 ounces. Given that he picked 15.5 times more oranges than pomegranates, how many of each of the three fruits did William pick?

So what we need to do in this question is form a series of equations. And these equations are gonna help us to find out the number of tree fruits that William picked. So how many lemons, oranges, and pomegranates.

So what I’ve done first of all is I’ve called lemons 𝑥, oranges 𝑦, and pomegranates 𝑧. So that’s the number of each of those fruit. Well, first of all, if we search for information, we can see that William picked 142 pieces of fruit. So therefore, our first equation can be 𝑥 plus 𝑦 plus 𝑧 is equal to 142. Because it’s the number of lemons plus the number of oranges plus the number of pomegranates equals 142.

Well, the next bit of information we’ve been given is that the total weight of the fruits is 70 pounds and 10 ounces. Well, as everything else is in ounces, what we want to do is convert this into ounces. Well, we know that one pound is equal to 16 ounces. So therefore, 70 pounds and 10 ounces is gonna be equal to 70 multiplied by 16 add 10, which is gonna be equal to 1130 ounces.

Okay, great. So let’s use this information to form another equation. And the other equation we can form is five 𝑥 plus eight 𝑦 plus 11𝑧 equals 1130. And we can do this cause we’re told that an orange weighs eight ounces, a lemon five ounces, and a pomegranate 11 ounces.

But do be careful because if we look at the beginning of the question, we can see that the order was lemon, orange, pomegranate. However, when we’re talking about the weights, it’s orange, lemon, pomegranate. So make sure they get in the right order. Otherwise, it’ll form the wrong equation for our second equation.

And the third equation we’re gonna get is 𝑦 equals 15.5𝑧. And we get that because we’re told that William picked 15.5 times more oranges than pomegranates. And 𝑦 represents the oranges and 𝑧 the pomegranates. Okay, great. So now what’s the next step?

Well, what we want to do is we want to combine a couple of our equations. And what we’re gonna want to do is substitute equation three into equation one. And that’s because we’ve got 𝑦 in terms of 𝑧. So now what we want to do is find 𝑥 in terms of 𝑧. And when we do that, we’re gonna get 𝑥 plus 15.5𝑧 plus 𝑧 equals 142. So then if we subtract 16.5𝑧 from each side of the equation. And we can do that because 15.5𝑧 plus 𝑧 is 16.5𝑧. Then what we’re gonna be left with is 𝑥 is equal to 142 minus 16.5𝑧.

So now we can see that we’ve got both 𝑦 and 𝑥 in terms of 𝑧. So what we can do is substitute both of these into equation two. So when we substitute these into equation two, what we’re gonna get is five multiplied by 142 minus 16.5𝑧 plus eight multiplied by 15.5𝑧 plus 11𝑧 is equal to 1130.

So now when we distribute across the parentheses, what we’re gonna get is 710 minus 82.5𝑧. That’s cause five multiplied by 142 is 710 and five multiplied by 16.5 is 82.5. Then plus 124𝑧 plus 11𝑧 equals 1130. So if we simplify this, we’re gonna get 710 plus 52.5𝑧 equals 1130.

So next, we can subtract 710 from each side of the equation. And this is gonna give us 52.5𝑧 equals 420. Then we divide both sides by 15.5 gives us 𝑧 is equal to eight. And that means that we know that William picked eight pomegranates.

So now what we can do first of all is find out what 𝑦 is. And we can find that out by substituting 𝑧 equals eight into equation three. Because we know now that 𝑦 is equal to 15.5 multiplied by eight, which is equal to 124. So now we found 𝑦. So that was the number of oranges.

What we can move on to now is the value for 𝑥, which is our number of lemons. So if we substitute 𝑧 equals eight into 𝑥 equals 142 minus 16.5𝑧, we’re gonna get 𝑥 is equal to 142 minus 16.5 multiplied by eight. So therefore, we’re gonna get 𝑥 is equal to 142 minus 132, which gives us 𝑥 is equal to 10. So therefore, we know that 10 lemons were picked.

So therefore we can say, in answer to the question “How many each of the three fruits did William pick?” We can say that William picked 10 lemons, 124 oranges, and eight pomegranates.