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Difficulty: MediumSimultaneous Equations and Systems

A solar micro-grid facility utilizes two distinct types of energy storage battery modules: Type-X and Type-Y. On Monday, a configuration of 4 Type-X modules and 3 Type-Y modules stored a total of 180 kWh180\text{ kWh} of energy. On Tuesday, a configuration of 2 Type-X modules and 5 Type-Y modules stored a total of 160 kWh160\text{ kWh} of energy. Assuming each battery module of a given type stores a constant amount of energy, what is the individual storage capacity of one Type-X module and one Type-Y module, respectively?

  1. Type-X = 30 kWh30\text{ kWh}, Type-Y = 20 kWh20\text{ kWh}Answer
  2. B
    Type-X = 20 kWh20\text{ kWh}, Type-Y = 30 kWh30\text{ kWh}
  3. C
    Type-X = 27 kWh27\text{ kWh}, Type-Y = 24 kWh24\text{ kWh}
  4. D
    Type-X = 25 kWh25\text{ kWh}, Type-Y = 25 kWh25\text{ kWh}
  5. E
    Type-X = 35 kWh35\text{ kWh}, Type-Y = 15 kWh15\text{ kWh}

Answer

Type-X = 30 kWh30\text{ kWh} and Type-Y = 20 kWh20\text{ kWh}
The correct option correctly states that Type-X has a capacity of 30 kWh30\text{ kWh} and Type-Y has a capacity of 20 kWh20\text{ kWh}. Substituting these values into both equations yields 4(30)+3(20)=1804(30) + 3(20) = 180 and 2(30)+5(20)=1602(30) + 5(20) = 160, confirming both daily constraints are satisfied.

Step-by-Step Solution

1
Formulate a system of two linear equations in two variables.
Let xx represent the capacity of one Type-X module and yy represent the capacity of one Type-Y module. The given conditions translate to:
Equation (1): 4x+3y=1804x + 3y = 180
Equation (2): 2x+5y=1602x + 5y = 160
Simultaneous word problems require establishing algebraic equations representing total quantities for each scenario.
2
Eliminate variable xx by multiplying Equation (2) by 2.
Equation (2) multiplied by 2 becomes:
Equation (3): 4x+10y=3204x + 10y = 320
Aligning coefficients of xx enables linear elimination.
3
Subtract Equation (1) from Equation (3) to solve for yy.
(4x+10y)−(4x+3y)=320−180  ⟹  7y=140  ⟹  y=20(4x + 10y) - (4x + 3y) = 320 - 180 \implies 7y = 140 \implies y = 20
Subtracting the equations eliminates xx, producing a single variable equation in yy.
4
Substitute y=20y = 20 back into Equation (1) to find xx.
4x+3(20)=180  ⟹  4x+60=180  ⟹  4x=120  ⟹  x=304x + 3(20) = 180 \implies 4x + 60 = 180 \implies 4x = 120 \implies x = 30
Determining the remaining unknown variable.

Key Concept

Solving Systems of Linear Equations in Two Variables
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Topics

Simultaneous Equations and Systems

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