Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Two students in Mr. Mitchell's class, Matt and Ruth, have been assigned a workbook to complete at their own pace. They get together at Matt's house after school to complete as many pages as they can. Matt has already completed $81$ pages and will continue working at a rate of $1$ page per hour. Ruth has completed $59$ pages and can work at a rate of $12$ 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, Matt and Ruth will have each completed $\_$ pages in their workbooks.

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Solve a system of equations using substitution: word problems

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

\newline

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

81

pages and will continue working at a rate of

1

page per hour. Ruth has completed

59

pages and can work at a rate of

12

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, Matt and Ruth will have each completed

\_

pages in their workbooks.

Define Variables: Let's define the variables: $\newline$ Let $x$ represent the number of hours it will take for Matt and Ruth to be on the same page. $\newline$ Let $M$ represent the total number of pages Matt will have completed. $\newline$ Let $R$ represent the total number of pages Ruth will have completed. $\newline$ We can write two equations to represent the situation: $\newline$ For Matt: $M = 81 + 1 \times x$ (since Matt has already completed $81$ pages and works at a rate of $1$ page per hour). $\newline$ For Ruth: $R = 59 + 12 \times x$ (since Ruth has already completed $59$ pages and works at a rate of $12$ pages per hour). $\newline$ Since they will eventually be working on the same page, we set $M$ equal to $R$ : $\newline$ $M$ $1$
Write Equations: Now we solve for $x$ using substitution: $\newline$ $81 + x = 59 + 12x$ $\newline$ Subtract $x$ from both sides: $\newline$ $81 = 59 + 11x$ $\newline$ Subtract $59$ from both sides: $\newline$ $22 = 11x$ $\newline$ Divide both sides by $11$ : $\newline$ $x = 2$
Solve for x: Now that we have the value of $x$ , we can find out how many pages each student will have completed after $2$ hours: $\newline$ For Matt: $M = 81 + 1 \times 2 = 81 + 2 = 83$ pages $\newline$ For Ruth: $R = 59 + 12 \times 2 = 59 + 24 = 83$ pages