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Difficulty: Very hardSystems of Linear Equations

An electronics manufacturer produces three types of circuit boards: Alpha, Beta, and Gamma. Production requires processing across three specialized workstations: Solder, Component Placement, and Inspection.

- Each Alpha board requires 2 hours of Solder, 3 hours of Component Placement, and 1 hour of Inspection.
- Each Beta board requires 1 hour of Solder, 4 hours of Component Placement, and 2 hours of Inspection.
- Each Gamma board requires 3 hours of Solder, 2 hours of Component Placement, and 4 hours of Inspection.

During a given production cycle, the Solder station was operated for 55 hours, the Component Placement station for 85 hours, and the Inspection station for 65 hours. If all three workstations were operated at full capacity with no downtime, what was the total number of circuit boards produced?

Answer: 28 circuit boards

Answer

The total number of circuit boards produced is 28.
Setting up equations for total machine hours yields 2x+y+3z=552x + y + 3z = 55, 3x+4y+2z=853x + 4y + 2z = 85, and x+2y+4z=65x + 2y + 4z = 65, where xx, yy, and zz represent the quantities of Alpha, Beta, and Gamma boards produced, respectively. Subtracting the third equation from the first equation gives (2x+y+3z)−(x+2y+4z)=55−65(2x + y + 3z) - (x + 2y + 4z) = 55 - 65, which simplifies to x−y−z=−10x - y - z = -10, or x=y+z−10x = y + z - 10. Substituting x=y+z−10x = y + z - 10 into the second and third equations produces the 2x2 system 7y+5z=1157y + 5z = 115 and 3y+5z=753y + 5z = 75. Subtracting these two equations eliminates zz, giving 4y=404y = 40, so y=10y = 10. Substituting y=10y = 10 into 3y+5z=753y + 5z = 75 gives 30+5z=7530 + 5z = 75, so z=9z = 9. Substituting y=10y = 10 and z=9z = 9 into x=y+z−10x = y + z - 10 gives x=9x = 9. The total number of circuit boards produced is x+y+z=9+10+9=28x + y + z = 9 + 10 + 9 = 28.

Step-by-Step Solution

1
Set up the 3x3 system of linear equations based on workstation hours.
2x+y+3z=552x + y + 3z = 55, 3x+4y+2z=853x + 4y + 2z = 85, and x+2y+4z=65x + 2y + 4z = 65
Each equation models the total operational hours used across the three product types.
2
Subtract the third equation from the first equation to isolate xx in terms of yy and zz.
x=y+z−10x = y + z - 10
Eliminating terms directly reduces coefficient complexity.
3
Substitute x=y+z−10x = y + z - 10 into the second and third equations to construct a 2x2 system.
3y+5z=753y + 5z = 75 and 7y+5z=1157y + 5z = 115
Reducing to a two-variable system allows direct elimination.
4
Subtract the two reduced equations to solve for yy and zz.
y=10y = 10 and z=9z = 9
The 5z5z terms cancel out upon subtraction.
5
Determine xx and compute the total sum x+y+zx + y + z.
x=9x = 9, total =9+10+9=28= 9 + 10 + 9 = 28
The question asks for the total quantity of circuit boards produced.

Key Concept

Systems of Linear Equations
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Topics

Systems of Linear Equations

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