Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Last Wednesday, two friends met up after school to read the book they were both assigned in Literature class. Maya can read $2$ pages per minute, and she had already read $87$ pages. Ariel, who has a reading speed of $4$ pages per minute, had read $9$ pages. Eventually they had read the same number of pages. How long did that take? How many pages had each of them read? $\newline$ After $\_\_$ minutes, Maya and Ariel had each read $\_\_$ pages.

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Grade 8

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

Last Wednesday, two friends met up after school to read the book they were both assigned in Literature class. Maya can read

2

pages per minute, and she had already read

87

pages. Ariel, who has a reading speed of

4

pages per minute, had read

9

pages. Eventually they had read the same number of pages. How long did that take? How many pages had each of them read?

\newline

After

\_\_

minutes, Maya and Ariel had each read

\_\_

pages.

Define Variables: Let's define the variables: $\newline$ Let $t$ be the time in minutes after they started reading at the same time. $\newline$ Let $P_m$ be the total number of pages Maya has read after $t$ minutes. $\newline$ Let $P_a$ be the total number of pages Ariel has read after $t$ minutes. $\newline$ We can write two equations to represent the situation: $\newline$ For Maya: $P_m = 87 + 2t$ (since she reads $2$ pages per minute and had already read $87$ pages). $\newline$ For Ariel: $P_a = 9 + 4t$ (since she reads $4$ pages per minute and had already read $P_m$ $0$ pages). $\newline$ We want to find the value of $t$ when $P_m$ $2$ .
Set Up Equations: Now we set up the system of equations: $\newline$ $1$ ) $P_m = 87 + 2t$ $\newline$ $2$ ) $P_a = 9 + 4t$ $\newline$ Since $P_m = P_a$ , we can equate the two expressions: $\newline$ $87 + 2t = 9 + 4t$
Solve for t: Next, we solve for t using substitution: $\newline$ Subtract $2t$ from both sides of the equation: $\newline$ $87 + 2t - 2t = 9 + 4t - 2t$ $\newline$ This simplifies to: $\newline$ $87 = 9 + 2t$
Substitute and Solve: Now, subtract $9$ from both sides of the equation: $\newline$ $87 - 9 = 9 + 2t - 9$ $\newline$ This simplifies to: $\newline$ $78 = 2t$
Find Pages Read: Divide both sides by $2$ to solve for $t$ : $\frac{78}{2} = \frac{2t}{2}$ This gives us: $t = 39$
Find Pages Read: Divide both sides by $2$ to solve for $t$ : $\frac{78}{2} = \frac{2t}{2}$ This gives us: $t = 39$ Now that we have the value of $t$ , we can find out how many pages each of them read by substituting $t$ back into the original equations:For Maya: $P_m = 87 + 2t = 87 + 2(39) = 87 + 78 = 165$ pagesFor Ariel: $P_a = 9 + 4t = 9 + 4(39) = 9 + 156 = 165$ pages