The graph of \(3x + 2y = 12\) meets the \(y\)-axis at
- (a)\((4, 0)\)
- (b)\((0, 4)\)
- (c)\((6, 0)\)
- (d)\((0, 6)\)
Pair of Linear Equations in Two Variables Class 10: notes and important questions
Revision notes, 43 board-style questions with the step marks shown, and a four-step route from the basics to 95+, each step ending in a short checkpoint.
- 43 questions
- 12 multiple choice, 3 assertion–reason, 10 very short answer, 10 short answer, 5 long answer, 3 case study
- About 12 hours to master
Algebra unit: 20 of 80 theory marks (Polynomials, Pair of Linear Equations, Quadratic Equations, Arithmetic Progressions).
Revision notes
Pair of Linear Equations in Two Variables — revision notes
1. General form and graphs
\(a_1x + b_1y + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\). Each equation is a straight line; a solution is a point lying on both lines.
2. Number of solutions (consistency)
| Ratios | Lines | Solutions | Pair is |
|---|---|---|---|
| \(\dfrac{a_1}{a_2} \neq \dfrac{b_1}{b_2}\) | intersecting | exactly one | consistent |
| \(\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}\) | coincident | infinitely many | consistent (dependent) |
| \(\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2}\) | parallel | none | inconsistent |
3. Graphical method
- Make a table of at least 2–3 points for each line (intercepts are easiest), plot, and read the intersection point.
- Triangles formed with an axis: find where each line meets that axis, plus the intersection point; area \(= \tfrac12 \times \text{base} \times \text{height}\).
4. Algebraic methods
- Substitution: make one variable the subject of one equation and substitute into the other.
- Elimination: multiply to make the coefficients of one variable equal, then add/subtract.
- Symmetric coefficients (e.g. \(47x + 31y,\ 31x + 47y\)): add and subtract the equations to get \(x + y\) and \(x - y\).
Worked example 1
Solve \(4x + 3y = 18,\ 2x - y = 4\). From the second, \(y = 2x - 4\); then \(4x + 6x - 12 = 18 \Rightarrow x = 3,\ y = 2\).
Worked example 2
For which \(k\) is \(3x + ky = 9,\ 6x + 4y = 18\) dependent? \(\tfrac36 = \tfrac{k}{4} = \tfrac{9}{18} \Rightarrow k = 2\).
Worked example 3
A number of ₹10 and ₹20 notes total 30 notes worth ₹440: \(x + y = 30,\ 10x + 20y = 440 \Rightarrow y = 14,\ x = 16\).
Common errors
- Comparing only \(a\) and \(b\) ratios and forgetting \(c\) when deciding between parallel and coincident.
- Sign errors when subtracting equations; check the answer in both original equations.
- “Consistent” means at least one solution — not necessarily a unique one.
- In graphs: unlabelled axes/lines, or too few points plotted.
Board-exam tips
- Word problems: define your variables in words first (“let the cost of one pen be ₹\(x\)”) — this earns the method mark.
- If a method is specified (graphical/elimination/substitution), you must use it.
- Write the final answer in context with units.
Topics in this chapter: Consistency and number of solutions · Elimination method · Graphical method of solution · Substitution method · Situational problems.
Route to 95: four steps
Work through the steps in order. Take each checkpoint closed book, about 1.5 minutes per mark; pass at 80% to move on. Two misses in a row means going back one step.
Step 1
Secure the basics
You can solve a pair of equations by substitution or elimination and tell from the ratios whether lines meet, are parallel or coincide.
Step 2
Board standard
You can find k for a given number of solutions, solve by the graph, and turn a board-style word problem into two equations.
Step 3
Full marks on long answers
You can produce neat graphs with labelled lines and complete word-problem solutions that define variables and state the final answer.
Step 4
95+ stretch (HOTS)
You can handle consistency questions with two unknowns, clever combinations of the solution and the slightly unusual word problems that separate 95+ scripts.
Practice questions
Original questions in the board's styles. Multiple-choice answers are checked as you go; for written answers, compare your working with the step mark scheme and record your marks. 11 of the 43 are competency-based (case studies and questions set in a real-life situation): the "Competency-based" button shows just those.
The lines \(x = 3\) and \(y = -2\) intersect at
- (a)\((3, -2)\)
- (b)\((-2, 3)\)
- (c)\((3, 0)\)
- (d)\((0, -2)\)
The pair of equations \(2x + 3y = 7\) and \(4x + 6y = 14\) has
- (a)a unique solution
- (b)no solution
- (c)exactly two solutions
- (d)infinitely many solutions
Stretch yourself: original algebra problems at Intermediate level (ages 13 to 16), with hints and full solutions: Problem I117 · Problem I124 · Problem I27 · Problem I109. All 29 →
Where marks are lost in Pair of Linear Equations in Two Variables
- Deciding 'parallel' vs 'coincident' from a1/a2 and b1/b2 only. Fix: always compare c1/c2 as well and write all three ratios.
- Graphs with too few points, unlabelled axes or unnamed lines. Fix: make a table of at least 3 points per line, label each line with its equation and mark the intersection with its coordinates.
- Using elimination when the question says 'by substitution' (or graphically). Fix: if a method is named, use it; otherwise the answer marks can be lost.
- Word problems without 'Let the cost of one pen be ₹x…'. Fix: define both variables in words; this line earns the first M1.
- Sign errors when subtracting equations. Fix: check the solution in both original equations before writing the final answer.
- Triangle-with-axis questions: using the wrong base or forgetting the intercept points. Fix: list the three vertices, then area = ½ × base × height.
Exam strategy: how answers are step-marked · competency-based questions
Pair of Linear Equations in Two Variables in our sample papers
Once step 3 is passed, test the chapter inside a full timed paper on the 2026-27 pattern (original papers by us, not official CBSE papers).
- Sample paper 1 (Board standard): questions 3, 27
- Sample paper 2 (Board standard): questions 3, 22
- Sample paper 3 (Above board): questions 4, 27
- Sample paper 4 (Above board): questions 4, 27
- Sample paper 5 (95+ challenge): questions 3, 32, 38
Want it against the clock? Take a timed 30-mark chapter test on Pair of Linear Equations in Two Variables, new questions each time, or climb the Difficulty ladder, levels 1 to 10, three questions a level (CBSE Essentials or the free trial).