You are fundraising for the schools track team by selling clothing. The first day, you sell 13 t-shirts and 17 pairs of shorts for a total of $385. The second day, you sell 13 t-shirts and 12 pairs of shorts for a total of $310. How much does a t-shirt cost and how much does a pair of shorts cost?

Answers:

one t-shirt costs  $10

one pair of shorts costs  $15

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Explanation:

x = cost of one t-shirt

y = cost of one pair of shorts

13x+17y = 385 is the first equation because 13x represents the money you get from selling 13 shirts, and the 17y represents the amount you get from selling 17 shorts. The total 13x+17y is given to be $385

The second equation is 13x+12y = 310 for similar reasoning, but this time you sell 12 shorts this time and you get $310 in revenue.

The system of equations we have is:

There are a few ways to solve this system. Possibly the quickest method is to note that the 13x shows up twice, so we can subtract straight down to eliminate the x terms

So,

After those three subtractions, we have the equation 5y = 75 which solves to y = 15 after dividing both sides by 5. This means the cost of one pair of shorts is $15

Plug this y value into any equation involving x and y, and solve for x

13x+17y = 385

13x+17(15) = 385

13x+255 = 385

13x = 385-255

13x = 130

x = 130/13

x = 10

Therefore, each t-shirt costs $10

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To help verify the answer, plug (x,y) = (10,15) into each equation. We should end up with a true result after simplifying both sides.

Start with the first equation

13x+17y = 385

13(10)+17(15) = 385

130+255 = 385

385 = 385

We get a true result which confirms the first equation.

Repeat for the second equation.

13x+12y = 310

13(10)+12(15) = 310

130+180 = 310

310 = 310

Both equations are true so (x,y) = (10, 15) has been fully confirmed.