Concept Topic GuideCBSE & ICSEClass 10MathematicsPair of Linear Equations in Two VariablesVerified Faculty Notes & Solutions
Pair of Linear Equations in Two Variables – Class 10 Maths Chapter 3
Class 10 Linear Equations notes: graphical and algebraic methods, substitution, elimination, cross-multiplication for CBSE & ICSE boards.
7 students readUpdated 4 September 2026
Chapter 3: Pair of Linear Equations in Two Variables
3.1 General Form
A pair of linear equations in two variables x and y:
a₁x + b₁y + c₁ = 0
a₂x + b₂y + c₂ = 0
a₂x + b₂y + c₂ = 0
3.2 Conditions for Consistency
| Condition | Type | Graphical Meaning | Solutions |
|---|---|---|---|
| a₁/a₂ ≠ b₁/b₂ | Consistent | Intersecting lines | Unique solution |
| a₁/a₂ = b₁/b₂ = c₁/c₂ | Dependent (Consistent) | Coincident lines | Infinitely many |
| a₁/a₂ = b₁/b₂ ≠ c₁/c₂ | Inconsistent | Parallel lines | No solution |
3.3 Algebraic Methods of Solving
Method 1: Substitution
- Express one variable in terms of the other from one equation.
- Substitute this into the second equation to get a single-variable equation.
- Solve and back-substitute.
Example: Solve x + y = 14 and x − y = 4
From equation 1: x = 14 − y. Substituting into equation 2: (14 − y) − y = 4 → 14 − 2y = 4 → y = 5, x = 9.
Method 2: Elimination
- Multiply equations to make coefficients of one variable equal.
- Add or subtract the equations to eliminate that variable.
- Solve the resulting single-variable equation.
Method 3: Cross-Multiplication
For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0:
x / (b₁c₂ − b₂c₁) = y / (c₁a₂ − c₂a₁) = 1 / (a₁b₂ − a₂b₁)
x / (b₁c₂ − b₂c₁) = y / (c₁a₂ − c₂a₁) = 1 / (a₁b₂ − a₂b₁)
3.4 Word Problems — Strategy
Steps for solving word problems:
1. Read the problem carefully and identify the unknowns — assign variables (x, y).
2. Translate the given conditions into two linear equations.
3. Solve using any algebraic method.
4. Verify the solution makes sense in the context of the problem.
1. Read the problem carefully and identify the unknowns — assign variables (x, y).
2. Translate the given conditions into two linear equations.
3. Solve using any algebraic method.
4. Verify the solution makes sense in the context of the problem.
Indexed Topics & Examination Keywords
#linear equations class 10#substitution method#elimination method#cross multiplication#consistent inconsistent#NCERT maths#CBSE#ICSE#simultaneous equations
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