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. Ronald can read $1$ page per minute, while Eve can read $2$ pages per minute. When they met, Ronald had already read $40$ pages, and Eve had already gotten through $39$ 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? $\newline$ Ronald and Eve 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. Ronald can read

1

page per minute, while Eve can read

2

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

40

pages, and Eve had already gotten through

39

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

Ronald and Eve had each read

\_

pages after

\_

minutes.

Define Variables: Define the variables for the system of equations. $\newline$ Let $x$ represent the number of minutes they read together, and $y$ represent the total number of pages read by each person.
Write Ronald's Equation: Write the equation for Ronald. $\newline$ Ronald's reading rate is $1$ page per minute, and he starts with $40$ pages already read. His equation is: $\newline$ $y = x + 40$
Write Eve's Equation: Write the equation for Eve. $\newline$ Eve's reading rate is $2$ pages per minute, and she starts with $39$ pages already read. Her equation is: $\newline$ $y = 2x + 39$
Set Up System: Set up the system of equations. $\newline$ The system of equations is: $\newline$ $y = x + 40$ $\newline$ $y = 2x + 39$
Solve Using Substitution: Solve the system using substitution. $\newline$ Since both equations are equal to $y$ , set them equal to each other to find $x$ : $\newline$ $x + 40 = 2x + 39$
Solve for x: Solve for x. $\newline$ Subtract $x$ from both sides of the equation: $\newline$ $x + 40 - x = 2x + 39 - x$ $\newline$ $40 = x + 39$ $\newline$ Now subtract $39$ from both sides: $\newline$ $40 - 39 = x + 39 - 39$ $\newline$ $1 = x$
Find y Value: Find the value of y by substituting x into one of the original equations. $\newline$ Using Ronald's equation: $\newline$ $y = x + 40$ $\newline$ $y = 1 + 40$ $\newline$ $y = 41$
Verify Solution: Verify the solution by substituting $x$ into Eve's equation. $y = 2x + 39$ $y = 2(1) + 39$ $y = 2 + 39$ $y = 41$ Since the value of $y$ is the same for both equations, the solution is correct.