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Chapter 3 · NCERT Maths 041

Pair of Linear Equations

Two lines, one question: where do they meet? That meeting point is your solution — and sometimes they never meet, or meet everywhere. Drag the lines and see.

📈 4 topics⏱ ~45 min📝 20-question quiz
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Two lines, one solution

Drag the slope and intercept of each line. Where they cross is the solution. Make them parallel (no solution) or land them on top of each other (infinitely many).

line 1 (blue) & line 2 (purple)drag the sliders
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The four ideas in this chapter

A pair of linear equations is two equations in the same two variables, e.g. x + y = 10 and x − y = 4. A solution is a pair (x, y) that makes both equations true at once.

Each equation is a straight line, so the solution is the point where the two lines meet. That single geometric idea drives the whole chapter.

Three possibilities

Two lines can cross once (one solution), be parallel (no solution), or be the same line (infinitely many solutions). There is no other case.

Plot both lines on the same axes. Their intersection point is the solution — read off its (x, y).

  • Intersecting lines → exactly one solution (consistent).
  • Parallel lines → no solution (inconsistent).
  • Coincident lines → infinitely many solutions (consistent, dependent).

Use the grapher near the top — drag the sliders until the lines cross, then make them parallel and watch the solution disappear.

Common mistake: reading the wrong axis. The solution is the (x, y) of the intersection point, not where a line meets an axis.

Graphs are great for understanding, but algebra gives exact answers. Two reliable methods:

Substitution · y from one equation into the other

Solve x + y = 10 and x − y = 4.

  1. From the first: y = 10 − x.
  2. Substitute into the second: x − (10 − x) = 4 → 2x − 10 = 4 → 2x = 14.
  3. x = 7, then y = 10 − 7 = 3. Solution (7, 3).
Elimination · add/subtract to cancel a variable

Solve 2x + 3y = 13 and 2x − y = 5.

  1. Subtract the equations (the 2x cancels): (3y) − (−y) = 13 − 5 → 4y = 8 → y = 2.
  2. Put y = 2 into 2x − y = 5: 2x − 2 = 5 → 2x = 7 → x = 3.5.
  3. Solution (3.5, 2). Check in both equations ✓.

For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, compare the ratios of the coefficients to know the case instantly:

  • a₁/a₂ ≠ b₁/b₂ → intersecting → unique solution.
  • a₁/a₂ = b₁/b₂ ≠ c₁/c₂ → parallel → no solution.
  • a₁/a₂ = b₁/b₂ = c₁/c₂ → coincident → infinitely many.
Worked example · classify without graphing

Is 2x + 3y = 5, 4x + 6y = 10 consistent?

  1. a₁/a₂ = 2/4 = 1/2; b₁/b₂ = 3/6 = 1/2; c₁/c₂ = 5/10 = 1/2.
  2. All three are equal → the lines are the same line.
  3. So there are infinitely many solutions (consistent & dependent).
Common mistake: writing equations as a₁x + b₁y = c₁ but forgetting to move c to one side consistently before comparing ratios. Keep the same form for both.

Why this matters

Where you'll actually use this

"When does plan A beat plan B?" is a pair of linear equations in disguise — and you'll meet that question constantly: phone plans, taxis, salaries, mixtures.

Which plan is cheaper?

Plan A: a low monthly fee but high per-minute rate. Plan B: higher fee, lower rate. Each is a straight line of cost vs usage, and the point where they cross is the break-even — below it one plan wins, above it the other. Telecom, electricity and SaaS pricing all use exactly this.

Intersection = break-even
x L y L mix + =

Mixing & recipes

A chemist (or a chef) blends x litres of one solution with y litres of another to hit a target volume and a target strength. That's two conditions in two unknowns — a pair of linear equations — solved by elimination.

Two conditions, two unknowns
🚕 Taxi vs ride-share

Base fare plus per-km rate for each service — solve to find the distance where costs match.

💼 Salary + commission

Two job offers (fixed + commission) become two lines; the intersection is the sales level where they pay equally.

🏭 Supply & demand

Economists find market price where the supply line and demand line intersect.

🔒 More real-world applications

Taxi pricing, salary offers, supply & demand and more — each explained with a diagram. Free to unlock.

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Competency quiz

Modelled on CBSE's competency pattern — MCQ, assertion–reason and case-study items.

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Interactive grapher hand-built for trykarkedekho. Content from the rationalised NCERT Class 10 Maths syllabus (ncert.nic.in).

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