Chapter 3 · CBSE Class 10 Maths
Pair of Linear Equations in Two Variables
Two ways to learn — a crisp formula reference, or a story that makes the ratio test feel obvious.
Four scenes that build the ideas from scratch, step by step.
How this works
Four scenes, one thread: two linear equations are two lines, and there are only three ways two lines can relate to each other. Once you can predict which way without graphing, two algebraic shortcuts find the exact answer. Read the story, then tap "The Maths" for the formal result.
Three Ways Two Lines Can Meet
Take three different pairs of equations and imagine graphing each one. Pair A: x − 2y = 0 and 3x + 4y − 20 = 0. Pair B: 2x + 3y − 9 = 0 and 4x + 6y − 18 = 0. Pair C: x + 2y − 4 = 0 and 2x + 4y − 12 = 0.
Graph all three pairs. You'll find each one behaves completely differently — and there are only three possible behaviours in total, ever.
Pair A: the two lines cross at exactly one point → ONE solution
Pair B: the two lines turn out to be the SAME line → INFINITE solutions
Pair C: the two lines run side-by-side, never touching → NO solution
Pair B is the most surprising one: 4x + 6y − 18 = 0 is just 2x + 3y − 9 = 0 multiplied by 2 — it was never a second, different line at all, just the same line written differently. Every point on it is a shared solution.
Reading the Answer Without Graphing
Graphing every pair of equations to find out which of the three cases you're in works, but it's slow — and error-prone if the solution has ugly coordinates. There's a shortcut: write both equations in the form a₁x + b₁y + c₁ = 0 and just compare three simple ratios.
Compare a₁/a₂, b₁/b₂ and c₁/c₂ for Pair A, B and C from Scene 1 — what pattern shows up?
Pair A: 1/3, −2/4 = −1/2 → a₁/a₂ ≠ b₁/b₂ (intersecting, matches!)
Pair B: 2/4=1/2, 3/6=1/2, −9/−18=1/2 → ALL THREE equal (coincident, matches!)
Pair C: 1/2, 2/4=1/2, −4/−12=1/3 → a₁/a₂ = b₁/b₂ but c₁/c₂ different (parallel, matches!)
Each ratio pattern predicted the exact graphical behaviour from Scene 1 — without drawing a single line. The logic: a₁/a₂ = b₁/b₂ means the two lines have the same slope (parallel-or-identical); adding c₁/c₂ into the comparison is what tells them apart.
Substitution — Aftab's Riddle
Aftab tells his daughter: "Seven years ago, I was seven times as old as you were then. Also, three years from now, I'll be three times as old as you will be." Two clues, two unknown ages — a classic pair-of-equations setup.
Let my age be t and yours be s. The first clue: s − 7 = 7(t − 7). The second: s + 3 = 3(t + 3). How do we untangle two unknowns from two clues?
Simplify both: s − 7t + 42 = 0 (1) s − 3t = 6 (2)
From (2): s = 3t + 6
Substitute into (1): (3t+6) − 7t + 42 = 0 → −4t + 48 = 0 → t = 12
Back into s = 3t+6: s = 3(12)+6 = 42
The trick, in general: solve ONE equation for one variable (here, s in terms of t), then plug that expression straight into the OTHER equation — collapsing two unknowns down to just one.
Elimination — and Its Dead Ends
Two people's incomes are in ratio 9:7, expenses in ratio 4:3, and each saves ₹2000/month. Call the incomes 9x and 7x, expenses 4y and 3y: 9x − 4y = 2000 and 7x − 3y = 2000. Neither variable is conveniently isolated here — substitution would get messy.
Instead of isolating a variable, scale BOTH equations until one variable's coefficient matches exactly — then just subtract to make it vanish.
×3 and ×4 to make the y-coefficient 12 in both:
27x − 12y = 6000 28x − 12y = 8000
Subtract: x = 2000 → substitute back: 9(2000) − 4y = 2000 → y = 4000
Incomes: 9(2000) = ₹18,000 and 7(2000) = ₹14,000. But elimination has a stranger side too — sometimes BOTH variables cancel, leaving only numbers.
Solve 2x + 3y = 8 and 4x + 6y = 7:
×2 the first: 4x + 6y = 16
Subtract the second: 0 = 9 ← a FALSE statement
"0 = 9" looks like a mistake, but it isn't — it's the complete answer. Since it's false for every x and y, the pair has NO solution (these lines are secretly parallel). If instead the leftover statement were something like "18 = 18" — always TRUE — that would mean infinitely many solutions instead (the equations were the same line).
The Big Picture
Every idea in this chapter branches from one picture: two equations are two lines, and lines can only cross once, never, or always (same line). The ratio test predicts which; substitution and elimination both find the exact point when one exists.
Predict the case without graphing?
Compare a₁/a₂, b₁/b₂, c₁/c₂
a₁/a₂ ≠ b₁/b₂ means?
Intersecting — one solution
All three ratios equal?
Coincident — infinite solutions
First two equal, third different?
Parallel — no solution
Both variables cancel — TRUE left over?
Infinite solutions (same line)
Both variables cancel — FALSE left over?
No solution (parallel lines)
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