Pair of Linear Equations in Two Variables Class 10 Notes (2026-27) — CBSE
Class 10 Maths Chapter 3 notes: graphical and algebraic solutions, consistency conditions, substitution and elimination methods, and word problems.
Pair of Linear Equations in Two Variables — Class 10 Maths Notes
Chapter Snapshot
A pair of linear equations in x and y is two straight-line equations solved together. This chapter shows how to find the solution graphically and algebraically (substitution and elimination), and how the ratios of the coefficients instantly tell you whether the pair has a unique solution, no solution, or infinitely many.
Board relevance: a scoring Algebra chapter. Expect a solving question (substitution or elimination), a consistency question, and a word problem.
Syllabus note (rationalised): the cross-multiplication method and equations reducible to a pair of linear equations have been removed. Use graphical, substitution, and elimination methods.
Key Concepts & Definitions
General form:
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Each equation is a straight line. Solving the pair means finding the point (x, y) that satisfies both — geometrically, where the two lines meet.
Three possibilities (graphical):
Lines Solution Called
Intersect at one point Unique solution Consistent
Parallel (never meet) No solution Inconsistent
Coincident (same line) Infinitely many solutions Consistent & dependent
Formulas
Consistency from coefficient ratios
Condition Lines Solution
a₁/a₂ ≠ b₁/b₂ Intersecting Unique solution (consistent)
a₁/a₂ = b₁/b₂ ≠ c₁/c₂ Parallel No solution (inconsistent)
a₁/a₂ = b₁/b₂ = c₁/c₂ Coincident Infinitely many (dependent)
Substitution method
1. From one equation, express one variable in terms of the other.
2. Substitute into the second equation → one equation in one variable.
3. Solve, then back-substitute for the other variable.
Elimination method
1. Multiply the equations to make the coefficient of one variable equal.
2. Add or subtract to eliminate that variable.
3. Solve for the remaining variable, then back-substitute.
Worked Examples
Example 1 — Elimination: Solve 2x + 3y = 11 and 2x − 4y = −24.
Subtract: (2x + 3y) − (2x − 4y) = 11 − (−24) → 7y = 35 → y = 5.
Put y = 5 in the first: 2x + 15 = 11 → x = −2. Solution: x = −2, y = 5.
Example 2 — Substitution: Solve x + 2y = −1 and 2x − 3y = 12.
From the first, x = −1 − 2y. Substitute: 2(−1 − 2y) − 3y = 12 → −2 − 7y = 12 → y = −2.
Then x = −1 − 2(−2) = 3. Solution: x = 3, y = −2.
Example 3 — Consistency: For what value of k do 2x + 3y = 7 and (k − 1)x + (k + 2)y = 3k have infinitely many solutions?
Need a₁/a₂ = b₁/b₂ = c₁/c₂: 2/(k−1) = 3/(k+2) = 7/3k.
From 2/(k−1) = 3/(k+2): 2(k+2) = 3(k−1) → 2k + 4 = 3k − 3 → k = 7. (Verify with the third ratio.) So k = 7.
Example 4 — Word problem: The sum of two numbers is 27 and their difference is 3. Find them.
x + y = 27, x − y = 3. Adding: 2x = 30 → x = 15; then y = 12. The numbers are 15 and 12.
Example 5 — Two-digit number: A two-digit number is 7 times the sum of its digits, and the number formed by reversing the digits is 18 less. Find the number.
Let the number be 10x + y. Condition 1: 10x + y = 7(x + y) → 3x = 6y → x = 2y. Condition 2: (10x + y) − (10y + x) = 18 → 9x − 9y = 18 → x − y = 2. Substituting x = 2y: 2y − y = 2 → y = 2, x = 4. The number is 42.
Example 6 — Speed of boat/stream: A boat goes 30 km upstream and 44 km downstream in 10 hours; it goes 40 km upstream and 55 km downstream in 13 hours. Setting u = speed upstream and v = speed downstream, 30/u + 44/v = 10 and 40/u + 55/v = 13. Treating 1/u and 1/v as variables turns this into a linear pair — solving gives u = 5 km/h and v = 11 km/h. Such "boat and stream" and "time and work" problems reduce to a pair of linear equations once you choose the right variables.
Important Question Patterns
1. Solve (2–3 marks): by substitution or elimination; reduce to one variable, solve, back-substitute.
2. Consistency (2–3 marks): use the ratio conditions to classify a pair, or find k for unique/no/infinitely many solutions.
3. Graphical (3 marks): plot both lines, read the intersection point; state whether the region is consistent.
4. Word problems (3–4 marks): two-digit numbers (10x + y), ages, fractions (numerator/denominator), speed of boat in still water and stream, cost of items — set up two equations and solve.
5. Special forms (2 marks): equations like a fraction problem or "x years ago / after n years" ages — translate carefully into two linear equations.
⚡ Quick Revision
- General form: a₁x + b₁y + c₁ = 0, a₂x + b₂y + c₂ = 0 — two straight lines.
- Consistency: a₁/a₂ ≠ b₁/b₂ → unique (intersecting); = = ≠ (a₁/a₂ = b₁/b₂ ≠ c₁/c₂) → no solution (parallel); all equal → infinitely many (coincident).
- Substitution: express one variable, plug into the other equation.
- Elimination: equalise a coefficient, add/subtract to remove a variable.
- Word problems: two-digit number = 10x + y (reverse = 10y + x); form two equations from the two given conditions.
- Removed from syllabus: cross-multiplication method and reducible equations.
Get Started Free | Features | Pricing | Blog