Textbook chapter 3
Represent two conditions using two equations. Decide whether both conditions have one common solution, infinitely many or none, and solve practical problems.
Textbook exercise solutions
- Exercise 3.1 — 7 question groups
- Exercise 3.2 — 3 question groups
- Exercise 3.3 — 2 question groups
Videos, notes and tests
English · Session 2026–27 · NCERT-aligned study guide
Chapter overview
Represent two conditions using two equations. Decide whether both conditions have one common solution, infinitely many or none, and solve practical problems.
Before you begin: Linear equations, plotting ordered pairs and signed arithmetic.
Key concepts · Solved examples · Practice · Mind map · Timed chapter test
Learn the key concepts
Graphical interpretation
A solution must lie on both lines. Intersecting lines have one solution; coincident lines have every point in common; parallel distinct lines have none. Plot at least two correct points for each line.
Consistency without unsafe ratios
For a₁x+b₁y+c₁=0 and a₂x+b₂y+c₂=0, a₁b₂−a₂b₁ ≠ 0 guarantees a unique solution. If it is zero, compare all coefficients to decide coincidence or distinct parallel lines. Determinants avoid dividing by a coefficient that may be zero.
Substitution and elimination
In substitution, isolate one variable and insert its expression into the other equation. In elimination, multiply equations to align one coefficient, then add or subtract. Define variables and units before translating a word problem.
Formula and result checklist
- Unique solution when a₁b₂ − a₂b₁ ≠ 0.
- With defined ratios: a₁/a₂ = b₁/b₂ ≠ c₁/c₂ means no solution; all three equal means infinitely many.
Solved examples
Example 1
Solve x+y=13 and x−y=3.
Add the equations: 2x=16, so x=8. Substitute into x+y=13 to get y=5. Both original equations check.
Example 2
A notebook costs ₹x and a pen ₹y. Two notebooks and three pens cost ₹84; one notebook and two pens cost ₹48. Find the prices.
2x+3y=84 and x+2y=48. Double the second: 2x+4y=96. Subtract the first to obtain y=12. Then x=48−24=24.
Example 3
Classify 2x+4y=10 and x+2y=7.
Double the second: 2x+4y=14. The left sides match but the right sides differ. No pair can satisfy both; the lines are parallel and distinct.
Try it yourself
Work on paper before opening each answer. Show the reasoning and include units when needed.
1. Solve 3x+2y=18 and x+y=7.
Check answer 1
Subtract twice the second from the first: x=4; y=3.
2. Classify x−2y=4 and 3x−6y=12.
Check answer 2
Infinitely many solutions; the second equation is three times the first.
3. Two tickets cost ₹70 together. One costs ₹10 more than the other. Find each price.
Check answer 3
Let prices be x,y. x+y=70 and x−y=10 give ₹40 and ₹30.
Common mistakes to avoid
- An equation obtained by multiplying another is not a new independent condition.
- Check the ordered pair in both equations, not just one.
Chapter mind map
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Take the 6-question, 10-minute chapter practice test. Each correct answer earns 1 mark; there is no negative marking. See explanations after submission.
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Read the corresponding chapter in the official NCERT textbook, then complete its exercises on paper. These NSL explanations and original practice questions support your textbook study; they are not an exercise-by-exercise solution manual.
Official CBSE syllabus 2026–27 · All chapters · Previous chapter · Next chapter
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