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. Josiah can read $1$ page per minute, while Edward can read $4$ pages per minute. When they met, Josiah had already read $97$ pages, and Edward had already gotten through $13$ pages. After a while, they had both read the same number of pages. How long did that take? How many pages had each one read? $\newline$ After $\_$ minutes, Josiah and Edward had each read $\_$ pages.

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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. Josiah can read

1

page per minute, while Edward can read

4

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

97

pages, and Edward had already gotten through

13

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

\newline

After

\_

minutes, Josiah and Edward had each read

\_

pages.

Define Variables: Let's define two variables: let $t$ be the time in minutes after Josiah and Edward start reading together, and let $J$ be the total number of pages Josiah has read after $t$ minutes, and $E$ be the total number of pages Edward has read after $t$ minutes. We can write two equations to represent the situation: $\newline$ $1$ . Josiah's rate of reading is $1$ page per minute, and he had already read $97$ pages. So, the total number of pages he has read after $t$ minutes is $J = 97 + 1 \times t$ . $\newline$ $2$ . Edward's rate of reading is $4$ pages per minute, and he had already read $J$ $0$ pages. So, the total number of pages he has read after $t$ minutes is $J$ $2$ . $\newline$ Since they have read the same number of pages after $t$ minutes, we can set $J$ equal to $E$ : $\newline$ $J$ $6$
Write Equations: Now we solve the equation for $t$ : $\newline$ $97 + t = 13 + 4t$ $\newline$ Subtract $t$ from both sides: $\newline$ $97 = 13 + 3t$ $\newline$ Subtract $13$ from both sides: $\newline$ $84 = 3t$ $\newline$ Divide both sides by $3$ : $\newline$ $t = 28$ $\newline$ So, it took $28$ minutes for Josiah and Edward to have read the same number of pages.
Solve for t: Now we need to find out how many pages each one has read. We can substitute $t$ back into either of the original equations. Let's use Josiah's equation: $\newline$ $J = 97 + 1 \times t$ $\newline$ $J = 97 + 1 \times 28$ $\newline$ $J = 97 + 28$ $\newline$ $J = 125$ $\newline$ So, after $28$ minutes, Josiah has read $125$ pages.
Find Josiah's Pages: We can also check Edward's equation to ensure our solution is consistent: $\newline$ $E = 13 + 4 \times t$ $\newline$ $E = 13 + 4 \times 28$ $\newline$ $E = 13 + 112$ $\newline$ $E = 125$ $\newline$ So, after $28$ minutes, Edward has also read $125$ pages. This confirms our solution is correct.