# Video: Forming and Solving Systems of Linear Equations in Two Unknowns

A store clerk sold 60 pairs of sneakers. The high-tops sold for \$98.99 and the low-tops sold for \$129.99. If the receipts for the two types of sales totaled \$6404.40, how many of each type of sneaker were sold?

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### Video Transcript

A store clerk sold 60 pair of sneakers. The high-tops sold for 98 dollars and 99 cents and the low-tops sold for 129 dollars and 99 cents. If the receipts for the two types of sales totaled 6404 dollars and 40 cents, how many of each type of sneaker were sold?

We’ve been given a lot of information here. Let’s see if we can sort through it. A total of 60 pair of sneakers were sold. Two kinds of sneakers were sold: high-tops and low-tops. Let’s just stop here.

We need to define some variables. Let’s use an ℎ for the high-tops that were sold and an 𝑙 for some low-tops that were sold.

One thing that we know is that all the shoes that were sold added up to 60 pair. So if we add the high-tops that were sold and the low-tops that were sold, they would equal 60 pair: ℎ plus 𝑙 equals 60. We also know that the high-tops cost 98 dollars and 99 cents.

And that low- the low-tops cost 129 dollars and 99 cents, for a total amount sold 6404 dollars and 40 cents.

We can also write another equation. We can write an equation that says, the money spent on ℎ, the high-tops, plus the money spent on 𝑙, the low-tops, equals 6404 dollars and 40 cents. But what goes here? How much money was spent on high-tops?

The amount of money that high-tops cost times the number of high-tops that were purchased, that’s how much money was spent on high-tops, so we say 98 dollars and 99 cents times ℎ.

And we can do the same thing for the money spent on the low-tops, the money that the low-tops cost multiplied by the number of low-tops sold. So we say 129.99 times 𝑙.

Now we have two true equations about the same situation. We wanna know what the intersection of these two equations are, what makes both of these equations true. Let’s solve this problem through elimination.

Our first equation has a coefficient of 98 dollars and 99 cents for ℎ, so we want our second equation to have that same coefficient. Let’s multiply ℎ plus 𝑙 equals 60 by 98.99: 98.99 times ℎ, 98.99 times 𝑙, and 98.99 times 60, which equals 5939 and 40 cents.

Now what I have done is I’ve copied down our first equation exactly how it was written. And now we wanna subtract our second equation from the first equation. our ℎs are the same values, so they’ll cancel out.

Next we subtract 98 dollars and 99 cents from 129 dollars and 99 cents. That leaves us with 31𝑙. We subtract 5939 dollars and 40 cents from 6404 dollars and 40 cents. And we’re left with 465.

Remember that our goal is to figure out what ℎ and 𝑙 represent, so here we’ll need to get 𝑙 by itself. 31 is being multiplied by 𝑙. To move 31 to the other side, we’ll need to divide both sides by 31. This tells us that the pairs of low-tops sold were 15; 15 pairs of low-tops were sold.

Remember that a total of 60 pair of sneakers were sold, so we can plug our 15 in for our equation ℎ plus 𝑙 equals 60. The number of high-tops sold plus 15 equals 60. We subtract 15 from both sides, which tells us that ℎ, the number of high-tops sold, equals 45. High-top pairs: 45; low-top pairs: 15.