A landscaping company sells two types of soil mixtures: a basic mixture and a premium mixt… — SAT

A landscaping company sells two types of soil mixtures: a basic mixture and a premium mixture. Each bag of basic mixture contains $4$ pounds of compost and $8$ pounds of sand. Each bag of premium mixture contains $6$ pounds of compost and $5$ pounds of sand. A landscaper purchases a combination of bags containing a total of $46$ pounds of compost and $57$ pounds of sand. How many bags of premium mixture did the landscaper purchase?

Answer: 5 bags

Answer

The system of equations representing the scenario is

4b + 6p = 46

for compost and

8b + 5p = 57

for sand, where

b

represents the number of bags of basic mixture and

p

represents the number of bags of premium mixture. Multiplying the first equation by

2

gives

8b + 12p = 92

. Subtracting the second equation from this yields

(8b + 12p) - (8b + 5p) = 92 - 57

, which simplifies to

7p = 35

. Dividing by

7

gives

p = 5

. Thus, the landscaper purchased

5

bags of premium mixture.

Step-by-Step Solution

Define variables and write the system of equations based on the given context.

Let

b

represent the number of basic mixture bags and

p

represent the number of premium mixture bags. The system is:

\begin{aligned} 4b + 6p &= 46 \\ 8b + 5p &= 57 \end{aligned}

Translating the word problem into a system of linear equations is necessary to solve for the unknowns.

Multiply the first equation by

2

to facilitate the elimination method.

8b + 12p = 92

Aligning the coefficients of

b

allows us to eliminate

b

by subtracting the two equations.

Subtract the second equation from the new equation.

(8b + 12p) - (8b + 5p) = 92 - 57 \implies 7p = 35

This isolates the variable

p

by eliminating the variable

b

Solve for

p

p = 5

Dividing both sides of the equation by

7

gives the final number of premium mixture bags.

Key Concept

Solving systems of linear equations in context

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