A landscaping company uses the equation 12x + 18y = 240 to model the total cost, in dollar… — SAT

A landscaping company uses the equation $12x + 18y = 240$ to model the total cost, in dollars, of renting a wood chipper for $x$ hours and a soil aerator for $y$ hours. If the wood chipper was rented for 5 more hours than the soil aerator, and the company spent a total of $240$ on these rentals, for how many hours was the soil aerator rented?

A
5
6Cevap
C
10
D
11

Cevap

The soil aerator was rented for 6 hours.

The correct answer is 6. By writing the relationship between the rental hours as

x = y + 5

and substituting this expression into the cost equation, we obtain

12(y + 5) + 18y = 240

. Distributing the 12 results in

12y + 60 + 18y = 240

. Combining the like terms gives

30y + 60 = 240

. Subtracting 60 from both sides of the equation yields

30y = 180

. Dividing both sides by 30 results in

y = 6

. Therefore, the soil aerator was rented for 6 hours.

Adım Adım Çözüm

Express the relationship between the rental hours of the wood chipper (

x

) and the soil aerator (

y

) as an equation.

x = y + 5

The wood chipper was rented for 5 more hours than the soil aerator.

Substitute the expression for

x

into the cost equation

12x + 18y = 240

12(y + 5) + 18y = 240

Substitution reduces the equation to a single variable, allowing us to solve for

y

Distribute the 12 and combine like terms.

12y + 60 + 18y = 240 \implies 30y + 60 = 240

Simplification isolates the variable terms on one side.

Isolate

y

by subtracting 60 from both sides and then dividing by 30.

30y = 180 \implies y = 6

This calculation yields the rental hours for the soil aerator.

Anahtar Kavram

Solving a system of linear equations in two variables where one equation represents a budget constraint and the other relates the quantities of the two variables.

Soru

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Adım Adım Çözüm

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