Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ To keep in shape, Dillon exercises at a track near his home. He requires $50$ minutes to do $10$ laps running and $5$ laps walking. In contrast, he requires $47$ minutes to do $9$ laps running and $5$ laps walking. Assuming he maintains a consistent pace while running and while walking, how long does Dillon take to complete a lap? $\newline$ Dillon takes $\_$ minutes to run a lap and $\_$ minutes to walk a lap.

Full solution

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

\newline

To keep in shape, Dillon exercises at a track near his home. He requires

50

minutes to do

10

laps running and

5

laps walking. In contrast, he requires

47

minutes to do

9

laps running and

5

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

\newline

Dillon takes

\_

minutes to run a lap and

\_

minutes to walk a lap.

Define Lap Times: Let's denote the time it takes Dillon to run a lap as $r$ minutes and the time it takes to walk a lap as $w$ minutes. We are given that $10$ laps running and $5$ laps walking take $50$ minutes, which gives us the equation $10r + 5w = 50$ .
Form Equations: Similarly, we are given that $9$ laps running and $5$ laps walking take $47$ minutes, which gives us the equation $9r + 5w = 47$ .
Eliminate Variable: We now have a system of two equations. We need to eliminate one of the variables, $r$ or $w$ . We choose to eliminate $w$ because its coefficients are the same in both equations.
Solve for $r$ : To eliminate $w$ , we can subtract the second equation from the first equation. This gives us $10r + 5w - (9r + 5w) = 50 - 47$ , which simplifies to $r = 3$ .
Substitute and Solve: Now that we have the value for $r$ , we can substitute it back into one of the original equations to solve for $w$ . We'll use the first equation: $10(3) + 5w = 50$ , which simplifies to $30 + 5w = 50$ .
Final Results: Subtracting $30$ from both sides of the equation gives us $5w = 20$ , and dividing both sides by $5$ gives us $w = 4$ .
Final Results: Subtracting $30$ from both sides of the equation gives us $5w = 20$ , and dividing both sides by $5$ gives us $w = 4$ .Therefore, Dillon takes $3$ minutes to run a lap and $4$ minutes to walk a lap.