Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ To keep in shape, Danny exercises at a track near his home. He requires $34$ minutes to do $9$ laps running and $4$ laps walking. In contrast, he requires $40$ minutes to do $10$ laps running and $5$ laps walking. Assuming he maintains a consistent pace while running and while walking, how long does Danny take to complete a lap? $\newline$ Danny 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, Danny exercises at a track near his home. He requires

34

minutes to do

9

laps running and

4

laps walking. In contrast, he requires

40

minutes to do

10

laps running and

5

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

\newline

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

Define Equations: Let's denote the time Danny takes to run a lap as $r$ minutes and the time he takes to walk a lap as $w$ minutes. We can then write two equations based on the information given: $\newline$ For $9$ laps running and $4$ laps walking taking $34$ minutes: $9r + 4w = 34$ $\newline$ For $10$ laps running and $5$ laps walking taking $40$ minutes: $10r + 5w = 40$
Use Elimination Method: To use elimination, we want the coefficients of one of the variables to be the same (or opposites) in both equations. We can multiply the first equation by $5$ and the second equation by $4$ to get the coefficients of $w$ to match: $\newline$ $(9r + 4w) \times 5 = 34 \times 5$ $\newline$ $(10r + 5w) \times 4 = 40 \times 4$
Multiply Equations: After multiplying, we get the new system of equations: $\newline$ $45r + 20w = 170$ $\newline$ $40r + 20w = 160$
Eliminate Variable: Now we can subtract the second equation from the first to eliminate $w$ : $(45r + 20w) - (40r + 20w) = 170 - 160$
Solve for r: This simplifies to: $5r = 10$
Substitute $r$ : Dividing both sides by $5$ to solve for $r$ gives us: $\newline$ $r = \frac{10}{5}$ $\newline$ $r = 2$ $\newline$ So, Danny takes $2$ minutes to run a lap.
Solve for w: Now we can substitute $r = 2$ into one of the original equations to solve for $w$ . We'll use the first equation: $\newline$ $9r + 4w = 34$ $\newline$ $9(2) + 4w = 34$
Solve for w: Now we can substitute $r = 2$ into one of the original equations to solve for $w$ . We'll use the first equation: $\newline$ $9r + 4w = 34$ $\newline$ $9(2) + 4w = 34$ This simplifies to: $\newline$ $18 + 4w = 34$
Solve for w: Now we can substitute $r = 2$ into one of the original equations to solve for $w$ . We'll use the first equation: $\newline$ $9r + 4w = 34$ $\newline$ $9(2) + 4w = 34$ This simplifies to: $\newline$ $18 + 4w = 34$ Subtracting $18$ from both sides gives us: $\newline$ $4w = 34 - 18$ $\newline$ $4w = 16$
Solve for w: Now we can substitute $r = 2$ into one of the original equations to solve for $w$ . We'll use the first equation: $\newline$ $9r + 4w = 34$ $\newline$ $9(2) + 4w = 34$ This simplifies to: $\newline$ $18 + 4w = 34$ Subtracting $18$ from both sides gives us: $\newline$ $4w = 34 - 18$ $\newline$ $4w = 16$ Dividing both sides by $4$ to solve for $w$ gives us: $\newline$ $w$ $0$ $\newline$ $w$ $1$ $\newline$ So, Danny takes $4$ minutes to walk a lap.