Pair of Linear Equations in Two Variables — Mathematics Class 10 Notes (CBSE & HBSE)
Free NCERT Mathematics notes for Pair of Linear Equations in Two Variables (Class 10) on Siksha Sarovar, aligned to CBSE and Haryana Board (HBSE). This chapter is broken into 3 topics with clear explanations, formulas, solved examples and board-pattern practice — free to read, no sign-up required.
Board exam focus — Pair of Linear Equations in Two Variables (CBSE & HBSE)
Very important chapter — typically 8-10 marks in CBSE. Covers graphical and algebraic methods. HBSE favors substitution and cross-multiplication. Word problems are common in both boards.
Graphical Method and Consistency
Pair of Linear Equations
General form: a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Types of Solutions
| Condition | Graphical Representation | Type of System |
|---|---|---|
| a₁/a₂ ≠ b₁/b₂ | Lines intersect at one point | Consistent (unique solution) |
| a₁/a₂ = b₁/b₂ = c₁/c₂ | Lines coincide (same line) | Consistent (infinite solutions) |
| a₁/a₂ = b₁/b₂ ≠ c₁/c₂ | Lines are parallel (no meeting) | Inconsistent (no solution) |
How to Check Before Solving
For equations a₁x+b₁y=c₁ and a₂x+b₂y=c₂:
Step 1: Compute a₁/a₂, b₁/b₂, c₁/c₂ Step 2: Compare ratios
Example: 2x + 3y = 7 and 4x + 6y = 5
- a₁/a₂ = 2/4 = 1/2
- b₁/b₂ = 3/6 = 1/2
- c₁/c₂ = 7/5
- a₁/a₂ = b₁/b₂ ≠ c₁/c₂ → Parallel lines → No solution (Inconsistent)
Graphical Solution
For intersecting lines:
- Make a table of (x, y) values for each equation
- Plot points and draw both lines
- Intersection point = solution
Exam Tip: Graphical method is best for checking number of solutions. For exact values, use algebraic methods.
Substitution and Elimination Methods
Method 1: Substitution
- From one equation, express one variable in terms of the other
- Substitute in the second equation
- Solve the resulting single-variable equation
- Back-substitute to find the other variable
Example: Solve x + y = 14 and x - y = 4
- From Eq 1: x = 14 - y
- Substitute in Eq 2: (14-y) - y = 4 → 14 - 2y = 4 → y = 5
- x = 14 - 5 = 9
- Solution: (9, 5)
Method 2: Elimination
- Multiply equations to make coefficients of one variable equal
- Add or subtract equations to eliminate that variable
- Solve for remaining variable
- Back-substitute
Example: Solve 2x + 3y = 9 and 4x - y = 3
- Multiply Eq 2 by 3: 12x - 3y = 9
- Add to Eq 1: 14x = 18 → x = 9/7
- Substitute: 2(9/7) + 3y = 9 → y = 3/7 (Hmm, let me use cleaner numbers)
Better Example: 3x + 2y = 12 and x - 2y = 0
- Add: 4x = 12 → x = 3
- From Eq 2: y = 3/2
- Solution: (3, 3/2)
Method 3: Cross-Multiplication
For a₁x+b₁y+c₁=0 and a₂x+b₂y+c₂=0: $$\frac{x}{b_1c_2-b_2c_1} = \frac{y}{c_1a_2-c_2a_1} = \frac{1}{a_1b_2-a_2b_1}$$
Use when: HBSE asks for cross-multiplication method specifically.
Strategy: Elimination is fastest for most problems. Substitution is more intuitive. Cross-multiplication is useful when denominators are known.
Word Problems on Linear Equations
Setting Up Word Problems
Strategy:
- Identify the two unknown quantities → assign variables (let x = ..., y = ...)
- Form two equations from the given conditions
- Solve using any algebraic method
- Check the answer satisfies both original conditions
Common Problem Types
Type 1: Age Problems "Present ages of A and B sum to 25. Five years hence, A will be twice B."
- Let A = x, B = y
- x + y = 25 ... (1)
- x + 5 = 2(y + 5) → x - 2y = 5 ... (2)
- Solve: x = 18.33... (adjust numbers for whole numbers)
Type 2: Number Problems "Sum of two numbers is 100, difference is 28." → x+y=100, x-y=28
Type 3: Speed/Distance Problems "A boat travels 30 km downstream in 2 hours and 18 km upstream in 3 hours."
- Speed of boat in still water = b, speed of current = c
- Downstream: b+c = 15, Upstream: b-c = 6
- Solve: b = 10.5, c = 4.5
Type 4: Fraction Problems "A fraction becomes 2/3 if 2 added to numerator, 1/3 if 4 subtracted from denominator."
- Let fraction = x/y
- (x+2)/y = 2/3 → 3x-2y+6=0
- x/(y-4) = 1/3 → 3x-y+12=0 Wait — careful with signs!
Key Formulas for Word Problems
| Scenario | Equations |
|---|---|
| Speed problems | Distance = Speed × Time; Downstream: (b+c)t, Upstream: (b-c)t |
| Number digit | Tens digit=x, Units=y → Number=10x+y; Reversed=10y+x |
| Age | Present age ± years = future/past age |
| Mixture | x litres of conc₁ + y litres of conc₂ = total |
CBSE Pattern: Word problems (2-3 marks) are standard. Always define variables clearly, state both equations, solve, and verify.
Frequently asked questions
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Do these notes follow CBSE and HBSE?
Yes. The Pair of Linear Equations in Two Variables notes are NCERT-aligned and include guidance for both CBSE and Haryana Board (HBSE), with important questions and MCQs for revision.
What does the Pair of Linear Equations in Two Variables chapter cover?
Concept explanations, key formulas and definitions, fully solved examples and board-pattern practice questions for Pair of Linear Equations in Two Variables.