Pair of Linear Equations in Two Variables
Chapter 3 | Mathematics
Complete NCERT notes, graphical method, algebraic methods (substitution, elimination, cross-multiplication), consistency conditions, word problems, and key points for CBSE Class 10 Board Exam 2025-26.
📖 Introduction
An equation of the form ax + by + c = 0 where a, b, c are real numbers and a, b are not both zero, is called a linear equation in two variables (x and y).
A pair of linear equations in two variables is written as:
The solution of a pair of linear equations is the ordered pair (x, y) that satisfies both equations.
📊 Graphical Method of Solution
In the graphical method, we plot both equations on the coordinate plane. The point(s) where the two lines intersect represent the solution(s).
🟢 Case 1: Intersecting Lines (Unique Solution)
If a₁/a₂ ≠ b₁/b₂, the lines intersect at a unique point → Consistent system
Example: x + y = 5 and x - y = 1 intersect at (3, 2)
🟡 Case 2: Coincident Lines (Infinite Solutions)
If a₁/a₂ = b₁/b₂ = c₁/c₂, the lines coincide → Dependent system (infinitely many solutions)
Example: 2x + 4y = 8 and x + 2y = 4 are the same line
🔴 Case 3: Parallel Lines (No Solution)
If a₁/a₂ = b₁/b₂ ≠ c₁/c₂, the lines are parallel → Inconsistent system (no solution)
Example: x + y = 5 and x + y = 7 are parallel lines
💡 Graphical Insight: The solution is the intersection point of the two lines.
🔄 Substitution Method
Steps:
- From one equation, express one variable in terms of the other (e.g., x in terms of y).
- Substitute this expression into the second equation.
- Solve the resulting single-variable equation.
- Substitute back to find the other variable.
📝 Example
Solve: x + y = 14, x - y = 4
From first: x = 14 - y
Substitute: (14 - y) - y = 4 → 14 - 2y = 4 → -2y = -10 → y = 5
Then x = 14 - 5 = 9
∴ Solution: (9, 5)
➖ Elimination Method
Steps:
- Multiply one or both equations by suitable constants to make coefficients of one variable equal (or opposites).
- Add or subtract the equations to eliminate that variable.
- Solve for the remaining variable.
- Substitute back to find the other variable.
📝 Example
Solve: 3x + 4y = 10, 2x + 3y = 8
Multiply eq1 by 3: 9x + 12y = 30
Multiply eq2 by 4: 8x + 12y = 32
Subtract: (9x - 8x) + (12y - 12y) = 30 - 32 → x = -2
Substitute: 3(-2) + 4y = 10 → -6 + 4y = 10 → 4y = 16 → y = 4
∴ Solution: (-2, 4)
✖️ Cross-Multiplication Method
For equations: a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
📝 Example
2x + 3y - 8 = 0, 3x + 2y - 7 = 0
a₁=2, b₁=3, c₁=-8; a₂=3, b₂=2, c₂=-7
x = (3×-7 - 2×-8)/(2×2 - 3×3) = (-21 + 16)/(4 - 9) = (-5)/(-5) = 1
y = (-8×3 - -7×2)/(4 - 9) = (-24 + 14)/(-5) = (-10)/(-5) = 2
∴ Solution: (1, 2)
🎯 Consistency Conditions
| Condition | Nature of Lines | Number of Solutions | Consistency |
|---|
| a₁/a₂ ≠ b₁/b₂ | Intersecting | Unique solution | Consistent |
| a₁/a₂ = b₁/b₂ = c₁/c₂ | Coincident | Infinite solutions | Dependent (Consistent) |
| a₁/a₂ = b₁/b₂ ≠ c₁/c₂ | Parallel | No solution | Inconsistent |
💡 Remember:
• Unique Solution: a₁/a₂ ≠ b₁/b₂
• No Solution: a₁/a₂ = b₁/b₂ ≠ c₁/c₂
• Infinite Solutions: a₁/a₂ = b₁/b₂ = c₁/c₂
📝 NCERT Solved Examples (In-depth)
Example 1: Solve by Substitution
Solve: 2x + 3y = 11 and 2x - 4y = -24
From eq1: 2x = 11 - 3y → x = (11 - 3y)/2
Substitute in eq2: 2[(11 - 3y)/2] - 4y = -24 → 11 - 3y - 4y = -24 → 11 - 7y = -24 → -7y = -35 → y = 5
Then x = (11 - 15)/2 = -4/2 = -2
∴ Solution: (-2, 5)
Example 2: Solve by Elimination
Solve: 3x + 4y = 25 and 5x - 6y = -9
Multiply eq1 by 3: 9x + 12y = 75
Multiply eq2 by 2: 10x - 12y = -18
Add: 19x = 57 → x = 3
Substitute in eq1: 3(3) + 4y = 25 → 9 + 4y = 25 → 4y = 16 → y = 4
∴ Solution: (3, 4)
Example 3: Word Problem
The sum of two numbers is 35 and their difference is 13. Find the numbers.
Let numbers be x and y (x > y)
x + y = 35 ...(1)
x - y = 13 ...(2)
Adding (1) and (2): 2x = 48 → x = 24
Then y = 35 - 24 = 11
∴ Numbers are 24 and 11.
Example 4: Condition for Consistency
For what value of k does the system have a unique solution?
2x + 3y = 7 and (k + 1)x + (2k - 1)y = 4k + 1
For unique solution: a₁/a₂ ≠ b₁/b₂
2/(k+1) ≠ 3/(2k-1)
2(2k-1) ≠ 3(k+1)
4k - 2 ≠ 3k + 3 → 4k - 3k ≠ 3 + 2 → k ≠ 5
∴ k can be any real number except 5.
⭐ Key Points for Board Exam (Deep Research)
- Linear Equation: ax + by + c = 0, where a, b are not both zero.
- Solution: Ordered pair (x, y) satisfying both equations.
- Graphical Interpretation: Two lines can intersect (unique), coincide (infinite), or be parallel (no solution).
- Substitution Method: Best when one variable has coefficient 1 or -1.
- Elimination Method: Best when coefficients are manageable; multiply to make coefficients equal.
- Cross-Multiplication: Direct formula for solution; useful when coefficients are large.
- Consistency: Consistent system has at least one solution; inconsistent has none.
- Homogeneous System: If c₁ = c₂ = 0, system always has trivial solution (0,0).
- Determinant Method: Δ = a₁b₂ - a₂b₁; if Δ ≠ 0 → unique solution.
- Word Problems: Carefully translate conditions into equations using variables.