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Difficulty: EasySimultaneous Equations and Systems

A manufacturing plant produces two types of solar panels: Model X and Model Y. Each Model X panel requires 22 hours of assembly and 11 hour of quality testing. Each Model Y panel requires 33 hours of assembly and 22 hours of quality testing. During a single week, the assembly line logged a total of 140140 hours and the quality testing line logged 8080 hours, with both departments operating at full capacity. How many Model X solar panels were produced during the week?

  1. A
    2020
  2. B
    3030
  3. 4040Answer
  4. D
    5050
  5. E
    6060

Answer

4040 Model X solar panels were produced during the week.
Setting up the simultaneous linear equations for assembly time (2x+3y=1402x + 3y = 140) and testing time (x+2y=80x + 2y = 80) and solving for xx gives x=40x = 40, which accurately satisfies both constraints.

Step-by-Step Solution

1
Define variables and set up the system of linear equations based on resource constraints.
Let xx be the number of Model X panels and yy be the number of Model Y panels.
Assembly constraint: 2x+3y=1402x + 3y = 140
Quality testing constraint: x+2y=80x + 2y = 80
Each department's total logged hours must equal the sum of hours required by both panel types.
2
Express xx in terms of yy using the testing constraint equation.
x=80−2yx = 80 - 2y
Isolating xx allows substitution into the assembly equation.
3
Substitute x=80−2yx = 80 - 2y into the assembly equation and solve for yy.
2(80−2y)+3y=140160−4y+3y=140160−y=140y=202(80 - 2y) + 3y = 140 160 - 4y + 3y = 140 160 - y = 140 y = 20
Solving a single-variable equation determines the quantity of Model Y panels.
4
Substitute y=20y = 20 back to find xx.
x=80−2(20)=40x = 80 - 2(20) = 40
Evaluating xx determines the quantity of Model X panels requested in the question stem.

Key Concept

Solving simultaneous linear equations by substitution or elimination
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Topics

Simultaneous Equations and Systems

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