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Question

Difficulty: MediumSystems of Linear Equations

A logistics warehouse uses two sizes of shipping crates: small crates and large crates. A shipment containing 33 small crates and 55 large crates has a total weight of 4747 pounds. A second shipment containing 66 small crates and 22 large crates has a total weight of 3838 pounds. What is the total weight, in pounds, of a shipment containing 22 small crates and 33 large crates?

  1. A
    11
  2. B
    22
  3. C
    26
  4. 29Answer
  5. E
    33

Answer

29 pounds
By setting up the linear system 3s+5l=473s + 5l = 47 and 6s+2l=386s + 2l = 38, using elimination gives s=4s = 4 pounds for a small crate and l=7l = 7 pounds for a large crate. Substituting these values into the target expression 2s+3l2s + 3l yields 2(4)+3(7)=292(4) + 3(7) = 29 pounds.

Step-by-Step Solution

1
Set up a system of linear equations from the given context.
Let ss be the weight of a small crate and ll be the weight of a large crate. The equations are: (1) 3s+5l=473s + 5l = 47 and (2) 6s+2l=386s + 2l = 38.
Translate the physical constraints of each shipment into algebraic equations.
2
Solve for the variable ll by eliminating ss.
Multiply Equation (1) by 22 to get 6s+10l=946s + 10l = 94. Subtract Equation (2) (6s+2l=386s + 2l = 38) from this new equation: (6s+10l)−(6s+2l)=94−38  ⟹  8l=56  ⟹  l=7(6s + 10l) - (6s + 2l) = 94 - 38 \implies 8l = 56 \implies l = 7.
Align the coefficients of ss so that elimination via subtraction yields a single-variable linear equation for ll.
3
Substitute l=7l = 7 back into Equation (2) to solve for ss.
6s+2(7)=38  ⟹  6s+14=38  ⟹  6s=24  ⟹  s=46s + 2(7) = 38 \implies 6s + 14 = 38 \implies 6s = 24 \implies s = 4.
Determine the individual weight of a small crate.
4
Calculate the requested total weight for 22 small crates and 33 large crates.
2s+3l=2(4)+3(7)=8+21=292s + 3l = 2(4) + 3(7) = 8 + 21 = 29.
Substitute the individual values of ss and ll into the target expression.

Key Concept

Solving 2x2 Systems of Linear Equations by Elimination and Linear Combination Evaluation
Estimated Time:1m 30s
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Topics

Systems of Linear Equations

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