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. Sophie can read $1$ page per minute, while Quincy can read $4$ pages per minute. When they met, Sophie had already read $83$ pages, and Quincy had already gotten through $11$ 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? $\newline$ Sophie and Quincy had each read $\_$ pages after $\_$ minutes.

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

1

page per minute, while Quincy can read

4

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

83

pages, and Quincy had already gotten through

11

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

Sophie and Quincy had each read

\_

pages after

\_

minutes.

Define Variables: Let's define the variables for the system of equations. Let $x$ represent the number of minutes they read together, and let $S$ represent the total number of pages Sophie has read, and $Q$ represent the total number of pages Quincy has read. $\newline$ Sophie's rate is $1$ page per minute, and she had already read $83$ pages. So, her equation will be: $\newline$ $S = 1 \times x + 83$ $\newline$ Quincy's rate is $4$ pages per minute, and he had already read $11$ pages. So, his equation will be: $\newline$ $Q = 4 \times x + 11$ $\newline$ We are given that after a while, they had both read the same number of pages, which means $S = Q$ .
Set Up Equations: Now we set up the system of equations using the information we have: $\newline$ $1$ . $S = x + 83$ $\newline$ $2$ . $Q = 4x + 11$ $\newline$ And since $S = Q$ , we can write: $\newline$ $x + 83 = 4x + 11$
Solve for x: Next, we solve for x using substitution. We already have $S = Q$ , so we can substitute $S$ for $Q$ in the second equation: $\newline$ $x + 83 = 4x + 11$ $\newline$ Now, we solve for x by getting all the x terms on one side and the constants on the other: $\newline$ $x - 4x = 11 - 83$ $\newline$ $-3x = -72$
Find Total Time: Divide both sides by $-3$ to find the value of $x$ : $\frac{-3x}{-3} = \frac{-72}{-3}$ $x = 24$ This means they read together for $24$ minutes.
Find Sophie's Pages: Now that we have the value of $x$ , we can find out how many pages each person read. We'll substitute $x$ back into one of the original equations. Let's use Sophie's equation: $\newline$ $S = x + 83$ $\newline$ $S = 24 + 83$ $\newline$ $S = 107$ $\newline$ So, Sophie read $107$ pages in total.
Check Quincy's Pages: We can also check Quincy's total pages read to ensure our solution is consistent: $\newline$ $Q = 4x + 11$ $\newline$ $Q = 4 \times 24 + 11$ $\newline$ $Q = 96 + 11$ $\newline$ $Q = 107$ $\newline$ Quincy also read $107$ pages in total, which confirms our solution is correct.