Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. $\newline$ Two students in Mr. Terry's class, Britney and Francesca, have been assigned a workbook to complete at their own pace. They get together at Britney's house after school to complete as many pages as they can. Britney has already completed $13$ pages and will continue working at a rate of $11$ pages per hour. Francesca has completed $27$ pages and can work at a rate of $9$ pages per hour. Eventually, the two students will be working on the same page. How long will that take? How many pages will each of them have completed? $\newline$ After $\_$ hours, Britney and Francesca will have each completed $\_$ pages in their workbooks.

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Grade 8

Solve a system of equations using any method: word problems

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

\newline

Two students in Mr. Terry's class, Britney and Francesca, have been assigned a workbook to complete at their own pace. They get together at Britney's house after school to complete as many pages as they can. Britney has already completed

13

pages and will continue working at a rate of

11

pages per hour. Francesca has completed

27

pages and can work at a rate of

9

pages per hour. Eventually, the two students will be working on the same page. How long will that take? How many pages will each of them have completed?

\newline

After

\_

hours, Britney and Francesca will have each completed

\_

pages in their workbooks.

Define Variables: Let's define the variables: $\newline$ Let $t$ be the time in hours after Britney and Francesca start working at Britney's house. $\newline$ Let $B$ be the total number of pages Britney will have completed after $t$ hours. $\newline$ Let $F$ be the total number of pages Francesca will have completed after $t$ hours. $\newline$ We can write two equations to represent the situation: $\newline$ For Britney: $B = 13 + 11t$ (since she has already completed $13$ pages and works at a rate of $11$ pages per hour). $\newline$ For Francesca: $F = 27 + 9t$ (since she has already completed $27$ pages and works at a rate of $B$ $0$ pages per hour). $\newline$ We want to find the time $t$ when $B$ equals $F$ , which means Britney and Francesca will be working on the same page.
Set Up Equations: Now we set up the equations to find when $B$ equals $F$ : $13 + 11t = 27 + 9t$ To solve for $t$ , we need to get all the terms with $t$ on one side and the constant terms on the other side. Subtract $9t$ from both sides: $13 + 11t - 9t = 27 + 9t - 9t$ $13 + 2t = 27$
Solve for t: Next, we subtract $13$ from both sides to isolate the term with $t$ : $\newline$ $13 + 2t - 13 = 27 - 13$ $\newline$ $2t = 14$ $\newline$ Now we divide both sides by $2$ to solve for $t$ : $\newline$ $\frac{2t}{2} = \frac{14}{2}$ $\newline$ $t = 7$
Calculate Pages Completed: We have found that $t = 7$ hours. Now we need to calculate how many pages each student will have completed after $7$ hours. $\newline$ For Britney: $\newline$ $B = 13 + 11t$ $\newline$ $B = 13 + 11(7)$ $\newline$ $B = 13 + 77$ $\newline$ $B = 90$ $\newline$ For Francesca: $\newline$ $F = 27 + 9t$ $\newline$ $F = 27 + 9(7)$ $\newline$ $F = 27 + 63$ $\newline$ $F = 90$
Final Result: We have found that after $7$ hours, both Britney and Francesca will have completed $90$ pages each in their workbooks.