An athlete participates in a charity event where they earn money by swimming laps in a poo… — SAT

An athlete participates in a charity event where they earn money by swimming laps in a pool and running laps on a track. The athlete earns $s$ dollars for each lap swum and $r$ dollars for each lap run. Completing 10 laps in the pool and 15 laps on the track raises a total of $135. Completing 12 laps in the pool and 20 laps on the track raises a total of$ 176. What is the value of $r$ ?

Answer: 7 dollars

Answer

The value of

r

is 7.

The correct answer is 7. To find this, we translate the word problem into a system of two linear equations:

10s + 15r = 135

and

12s + 20r = 176

. Simplifying these equations by dividing by their common factors yields

2s + 3r = 27

and

3s + 5r = 44

. Multiplying the first equation by 3 gives

6s + 9r = 81

, and multiplying the second equation by 2 gives

6s + 10r = 88

. Subtracting the two equations eliminates

s

and gives

r = 7

Step-by-Step Solution

Write the system of linear equations representing the athlete's earnings.

The system is:

10s + 15r = 135

12s + 20r = 176

To represent the relationships between the number of laps completed and the total amount raised.

Simplify both equations to work with smaller coefficients.

Divide the first equation by 5:

2s + 3r = 27

Divide the second equation by 4:

3s + 5r = 44

Simplifying equations reduces the likelihood of arithmetic errors.

Eliminate the variable

s

to solve for

r

Multiply the first simplified equation by 3 and the second by 2:

3(2s + 3r) = 3(27) \implies 6s + 9r = 81

2(3s + 5r) = 2(44) \implies 6s + 10r = 88

Subtract the first equation from the second:

(6s + 10r) - (6s + 9r) = 88 - 81 \implies r = 7

Using the elimination method allows us to solve directly for the target variable,

r

Key Concept

Solving a system of linear equations in two variables using elimination or substitution.

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