Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Two classmates got together over the weekend to do their assigned History reading. Leo can read $1$ page per minute, while Shawna can read $3$ pages per minute. When they met, Leo had already read $75$ pages, and Shawna had already gotten through $19$ pages. After a while, they had both read the same number of pages. How many pages had each one read? How long did that take? $\newline$ Leo and Shawna had each read $\_$ pages after $\_$ minutes.

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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

Two classmates got together over the weekend to do their assigned History reading. Leo can read

1

page per minute, while Shawna can read

3

pages per minute. When they met, Leo had already read

75

pages, and Shawna had already gotten through

19

pages. After a while, they had both read the same number of pages. How many pages had each one read? How long did that take?

\newline

Leo and Shawna had each read

\_

pages after

\_

minutes.

Define Variables: Let's define the variables: $\newline$ Let $x$ be the number of minutes they read together. $\newline$ Let $L$ be the total number of pages Leo read. $\newline$ Let $S$ be the total number of pages Shawna read. $\newline$ We can write two equations based on the given information: $\newline$ For Leo: $L = 75 + 1 \times x$ (since Leo reads $1$ page per minute and had already read $75$ pages) $\newline$ For Shawna: $S = 19 + 3 \times x$ (since Shawna reads $3$ pages per minute and had already read $19$ pages) $\newline$ We are also told that after a while, they had both read the same number of pages, which gives us the equation: $\newline$ $L = S$
Write Equations: Now we can set up the system of equations: $\newline$ $1$ . $L = 75 + x$ $\newline$ $2$ . $S = 19 + 3x$ $\newline$ And since $L = S$ , we can substitute the expression for $L$ from the first equation into the second equation: $\newline$ $75 + x = 19 + 3x$
Set Up System: Next, we solve for $x$ : $75 + x = 19 + 3x$ Subtract $x$ from both sides: $75 = 19 + 2x$ Subtract $19$ from both sides: $56 = 2x$ Divide both sides by $2$ : $x = 28$
Solve for x: Now that we have the value of $x$ , we can find the total number of pages each person read. We'll substitute $x$ back into the equations for $L$ and $S$ :
For Leo: $L = 75 + x = 75 + 28 = 103$ pages
For Shawna: $S = 19 + 3x = 19 + 3\times28 = 19 + 84 = 103$ pages
Find Total Pages: We have found that both Leo and Shawna read $103$ pages each after $28$ minutes.