A logistics company uses two types of boxes, small and large, to ship items. A shipment of… — SAT

A logistics company uses two types of boxes, small and large, to ship items. A shipment of $8$ small boxes and $5$ large boxes has a total weight of $180$ pounds. A second shipment of $6$ small boxes and $10$ large boxes has a total weight of $260$ pounds. What is the weight, in pounds, of one large box?

Answer: 20 pounds

Answer

The weight of one large box is 20 pounds.

The correct weight of one large box is 20 pounds. By formulating the system of linear equations representing the total weight of each shipment (

8s + 5L = 180

and

6s + 10L = 260

), we can eliminate

L

by multiplying the first equation by 2, resulting in

16s + 10L = 360

. Subtracting the second equation from this gives

10s = 100

, which solves to

s = 10

. Substituting

s = 10

back into

8s + 5L = 180

yields

80 + 5L = 180

, which simplifies to

5L = 100

, meaning

L = 20

Step-by-Step Solution

Set up a system of two linear equations based on the shipments.

Let

s

be the weight of a small box and

L

be the weight of a large box. The system of equations is:

\begin{aligned} 8s + 5L &= 180 \\ 6s + 10L &= 260 \end{aligned}

Translating the verbal statements into mathematical symbols represents the problem systemically.

Multiply the first equation by 2 to align the coefficients of

L

16s + 10L = 360

This sets up the variable

L

to have the same coefficient in both equations, allowing for elimination.

Subtract the second equation from the modified first equation to solve for

s

(16s + 10L) - (6s + 10L) = 360 - 260

10s = 100

s = 10

Subtracting eliminates the variable

L

, leaving a single-variable linear equation to solve.

Substitute

s = 10

back into the first equation to solve for

L

8(10) + 5L = 180

80 + 5L = 180

5L = 100

L = 20

Substituting the value of the solved variable determines the value of the remaining variable.

Key Concept

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

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