Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ To keep in shape, Julie exercises at a track near her home. She requires $62$ minutes to do $4$ laps running and $10$ laps walking. In contrast, she requires $65$ minutes to do $5$ laps running and $10$ laps walking. Assuming she maintains a consistent pace while running and while walking, how long does Julie take to complete a lap? $\newline$ Julie takes _ minutes to run a lap and _ minutes to walk a lap.

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Grade 8

Solve a system of equations using elimination: word problems

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Q. Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.

\newline

To keep in shape, Julie exercises at a track near her home. She requires

62

minutes to do

4

laps running and

10

laps walking. In contrast, she requires

65

minutes to do

5

laps running and

10

laps walking. Assuming she maintains a consistent pace while running and while walking, how long does Julie take to complete a lap?

\newline

Julie takes _____ minutes to run a lap and _____ minutes to walk a lap.

Define variables: Let's define two variables: let $r$ be the time in minutes it takes Julie to run a lap, and $w$ be the time in minutes it takes Julie to walk a lap. We can then write two equations based on the information given: $\newline$ For the first scenario ( $4$ laps running and $10$ laps walking taking $62$ minutes): $\newline$ $4r + 10w = 62$ $\newline$ For the second scenario ( $5$ laps running and $10$ laps walking taking $65$ minutes): $\newline$ $5r + 10w = 65$
Write equations: Now we will use the elimination method to solve the system of equations. To eliminate one of the variables, we can multiply the first equation by $-5$ and the second equation by $4$ , so that when we add the two equations, the terms with $r$ will cancel out. $\newline$ Multiplying the first equation by $-5$ : $\newline$ $-5(4r + 10w) = -5(62)$ $\newline$ $-20r - 50w = -310$ $\newline$ Multiplying the second equation by $4$ : $\newline$ $4(5r + 10w) = 4(65)$ $\newline$ $20r + 40w = 260$
Elimination method: Next, we add the two resulting equations to eliminate $r$ : $\newline$ $(-20r - 50w) + (20r + 40w) = -310 + 260$ $\newline$ $-20r + 20r - 50w + 40w = -310 + 260$ $\newline$ $0r - 10w = -50$ $\newline$ Now we can solve for $w$ by dividing both sides by $-10$ : $\newline$ $\frac{-10w}{-10} = \frac{-50}{-10}$ $\newline$ $w = 5$ $\newline$ Julie takes $5$ minutes to walk a lap.
Add equations: Now that we have the value for $w$ , we can substitute it back into one of the original equations to find $r$ . We'll use the first equation: $\newline$ $4r + 10w = 62$ $\newline$ $4r + 10(5) = 62$ $\newline$ $4r + 50 = 62$ $\newline$ Subtract $50$ from both sides to solve for $r$ : $\newline$ $4r = 62 - 50$ $\newline$ $4r = 12$ $\newline$ Divide both sides by $4$ : $\newline$ $r$ $0$ $\newline$ $r$ $1$ $\newline$ Julie takes $r$ $2$ minutes to run a lap.