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Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

Two classmates, Jason and Carly, plan to meet in the computer lab to type up their research papers. Jason can type at a speed of 3 pages per hour, whereas Carly can type 2 pages per hour. So far, Jason already has 10 pages typed up, compared to Carly's 14 pages. Once they sit down and start typing together, the two students will reach the same page count before too long. How long will that take? What will the page count be?

After
hours, each student will have a page count of
pages.

1 answer

To model this situation, let's define the variables and set up a system of equations.

Let \( t \) be the time in hours that Jason and Carly will be typing together.

  1. Jason's Page Count: Jason starts with 10 pages and types at a speed of 3 pages per hour. After \( t \) hours, he will have: \[ J(t) = 10 + 3t \]

  2. Carly's Page Count: Carly starts with 14 pages and types at a speed of 2 pages per hour. After \( t \) hours, she will have: \[ C(t) = 14 + 2t \]

We need to find when their page counts will be equal, which gives us the equation: \[ 10 + 3t = 14 + 2t \]

Now, let's solve the equation:

  1. Subtract \( 2t \) from both sides: \[ 10 + 3t - 2t = 14 \] \[ 10 + t = 14 \]

  2. Subtract 10 from both sides: \[ t = 14 - 10 \] \[ t = 4 \]

Now that we know they will meet after \( t = 4 \) hours, we can calculate the page count for both students at that time.

Calculating the Page Count:

  • For Jason: \[ J(4) = 10 + 3 \cdot 4 = 10 + 12 = 22 \]

  • For Carly: \[ C(4) = 14 + 2 \cdot 4 = 14 + 8 = 22 \]

Both Jason and Carly will have 22 pages after 4 hours of typing together.

Filling in the blanks: After 4 hours, each student will have a page count of 22 pages.