Methods of Solving Simultaneous Equations & Word Problems
1. The Three Algebraic Solution Methods
Algebraic Methods for Solving Simultaneous Systems
Method 1
Elimination by Equating Coefficients:
Multiply one or both equations by suitable constants so that coefficients of one variable match, then add or subtract to eliminate that variable.
Method 2
Method of Substitution:
Express one variable in terms of the other from one equation, and substitute this expression into the second equation to produce a single-variable equation.
Method 3
Cross-Multiplication Method:
Direct formula using coefficient cross-products: x / (b₁c₂ - b₂c₁) = y / (c₁a₂ - c₂a₁) = 1 / (a₁b₂ - a₂b₁).
2. Solving Commercial Business Word Problems
Problem 1: Optimal Product Pricing
Question: A departmental store sells 3 shirts and 2 pairs of trousers for ₹3,500. Another customer buys 2 shirts and 4 pairs of trousers of the same quality for ₹5,000. Determine the individual cost of one shirt and one pair of trousers.
Step-by-Step Solution:
- Define the variables:
- Let cost of 1 shirt = x (in ₹)
- Let cost of 1 pair of trousers = y (in ₹)
- Formulate the system of equations:
- Equation (1):
3x + 2y = 3,500 - Equation (2):
2x + 4y = 5,000
- Equation (1):
- Simplify Equation (2) by dividing through by 2:
x + 2y = 2,500- Express x in terms of y:
x = 2,500 - 2y
- Substitute x into Equation (1):
3 × (2,500 - 2y) + 2y = 3,5007,500 - 6y + 2y = 3,5007,500 - 4y = 3,5004y = 7,500 - 3,500 = 4,000y = 4,000 ÷ 4 = 1,000
- Substitute y back to find x:
x = 2,500 - 2 × (1,000)x = 2,500 - 2,000 = 500
- Conclusion:
- Cost of one shirt = ₹500
- Cost of one pair of trousers = ₹1,000
- Verification: (3 × 500) + (2 × 1,000) = 1,500 + 2,000 = ₹3,500 (Matches!)
Problem 2: Fixed and Variable Cost Allocation
Question: The total operating cost of an industrial printing press is partly fixed and partly varies directly with the number of brochures printed. Printing 1,000 brochures costs ₹4,500, while printing 2,500 brochures costs ₹8,250. Find the fixed monthly overhead cost and the variable cost per brochure.
Step-by-Step Solution:
- Define variables:
- Fixed monthly cost = F (in ₹)
- Variable cost per brochure = v (in ₹)
- Formulate equations:
- For 1,000 brochures:
F + 1,000v = 4,500…(Equation 1) - For 2,500 brochures:
F + 2,500v = 8,250…(Equation 2)
- For 1,000 brochures:
- Subtract Equation (1) from Equation (2) to eliminate F:
(F - F) + (2,500v - 1,000v) = 8,250 - 4,5001,500v = 3,750v = 3,750 ÷ 1,500 = 2.50
- Substitute v = 2.50 into Equation (1):
F + 1,000 × (2.50) = 4,500F + 2,500 = 4,500F = 4,500 - 2,500 = 2,000
- Conclusion:
- Fixed monthly overhead = ₹2,000
- Variable cost per brochure = ₹2.50
- Verification: 2,000 + (2,500 × 2.50) = 2,000 + 6,250 = ₹8,250 (Matches!)