An electronics manufacturing plant produces two types of circuit modules: Module X and Mod… — GMAT

An electronics manufacturing plant produces two types of circuit modules: Module X and Module Y. Producing one Module X requires $3$ minutes on Machine A and $5$ minutes on Machine B. Producing one Module Y requires $4$ minutes on Machine A and $2$ minutes on Machine B. During a certain shift, Machine A was operated for a total of $230$ minutes and Machine B was operated for a total of $220$ minutes. If both machines operated at full capacity with no idle time, how many Module X units were produced?

Answer: 30 units

Answer

Formulating equations from the machine times gives

3x + 4y = 230

for Machine A and

5x + 2y = 220

for Machine B. Multiplying the second equation by

2

yields

10x + 4y = 440

. Subtracting

3x + 4y = 230

from

10x + 4y = 440

eliminates

y

and yields

7x = 210

, which solves to

x = 30

Step-by-Step Solution

Define variables for the unknown quantities.

Let

x

be the number of Module X units produced, and let

y

be the number of Module Y units produced.

Representing unknown quantities with algebraic variables allows for model formulation.

Formulate a system of linear equations from the machine time constraints.

Machine A equation:

3x + 4y = 230

Machine B equation:

5x + 2y = 220

The total operational time on each machine equals the sum of times spent producing each module type.

Multiply the Machine B equation by

2

to enable elimination of

y

2 \times (5x + 2y) = 2 \times 220 \implies 10x + 4y = 440

Matching the coefficient of

y

with the first equation (

4y

) allows elimination via subtraction.

Subtract the Machine A equation from the modified Machine B equation and solve for

x

(10x + 4y) - (3x + 4y) = 440 - 230 \implies 7x = 210 \implies x = 30

Subtracting eliminates

y

directly, leaving a single linear equation in terms of

x

Key Concept

Solving a system of two linear equations in two variables using elimination

Estimated Time:1m 30s

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