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Difficulty: Very hardSystems of Linear Equations

A logistics company ships three types of packages: Small, Medium, and Large. A shipment containing 33 Small, 55 Medium, and 22 Large packages has a total weight of 170170 kilograms. A second shipment containing 11 Small, 22 Medium, and 11 Large package has a total weight of 6565 kilograms. What is the total weight, in kilograms, of a shipment containing 55 Small, 99 Medium, and 44 Large packages?

  1. A
    235235
  2. B
    275275
  3. 300300Answer
  4. D
    335335
  5. E
    405405

Answer

The total weight of the shipment is 300300 kilograms.
The target quantity 5S+9M+4L5S + 9M + 4L can be expressed directly as 1×(3S+5M+2L)+2×(1S+2M+1L)1 \times (3S + 5M + 2L) + 2 \times (1S + 2M + 1L). Substituting the given total weights yields 1(170)+2(65)=3001(170) + 2(65) = 300 kilograms.

Step-by-Step Solution

1
Formulate linear equations representing the package weights.
Let SS, MM, and LL represent the weight of a Small, Medium, and Large package respectively. Equation 1: 3S+5M+2L=1703S + 5M + 2L = 170. Equation 2: 1S+2M+1L=651S + 2M + 1L = 65.
Setting up algebraic representations for the system based on the problem statement.
2
Determine if the requested expression 5S+9M+4L5S + 9M + 4L can be formed as a linear combination c1(3S+5M+2L)+c2(1S+2M+1L)c_1(3S + 5M + 2L) + c_2(1S + 2M + 1L).
Match coefficients: 3c1+c2=53c_1 + c_2 = 5, 5c1+2c2=95c_1 + 2c_2 = 9, and 2c1+c2=42c_1 + c_2 = 4.
Since individual values of SS, MM, and LL cannot be uniquely determined from two equations with three variables, we seek scalar constants c1c_1 and c2c_2.
3
Solve for the multipliers c1c_1 and c2c_2.
Subtracting 2c1+c2=42c_1 + c_2 = 4 from 3c1+c2=53c_1 + c_2 = 5 yields c1=1c_1 = 1. Substituting c1=1c_1 = 1 into 2c1+c2=42c_1 + c_2 = 4 gives c2=2c_2 = 2. Checking the middle equation: 5(1)+2(2)=95(1) + 2(2) = 9, which holds true.
Finding the scalar weights that recreate the exact combination requested.
4
Compute the total weight using the linear combination of the known values.
Total Weight =1×170+2×65=170+130=300= 1 \times 170 + 2 \times 65 = 170 + 130 = 300 kilograms.
Applying the solved linear combination to the total weights of the shipments.

Key Concept

Solving for a linear combination of variables in an underdetermined system without finding individual variable values.
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Topics

Systems of Linear Equations

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