Pair of Linear Equations: Two Clues, One Answer
Learn how a pair of linear equations in two variables works, the three types of solutions, and how to solve them by elimination with a step-by-step worked example.
By the PadhoDost Team · 📖 6 min read · Updated 4 August 2026
Part of Class 10 (CBSE) prep🧠 Two clues, one answer
Imagine a detective with two clues about a suspect: 'the two numbers add up to 10' and 'their difference is 4'. Neither clue alone pins down the numbers, but together they lock in exactly one pair (7 and 3). A pair of linear equations works the same way - two conditions that must both be true at the same time, meeting at a single answer.
What a pair of linear equations means
A linear equation in two variables looks like 2x + 3y = 13. On its own it has infinitely many solutions. The general form is a1x + b1y + c1 = 0, where a and b are not both zero. A 'pair' means two such equations taken together, and solving them means finding the one value of x and one value of y that satisfy BOTH equations at once. Graphically, each equation is a straight line, and the solution is the point where the two lines cross.
Three types of solutions
When you draw the two lines, only three things can happen: they cross at one point, they lie exactly on top of each other, or they stay parallel and never meet. You can predict which case you have just by comparing the ratios of the coefficients - no graph needed.
| Compare the ratios | The lines are | Number of solutions | Consistency |
|---|---|---|---|
| a1/a2 is not equal to b1/b2 | Intersecting | Exactly one (unique) | Consistent |
| a1/a2 = b1/b2, not equal to c1/c2 | Parallel | No solution | Inconsistent |
| a1/a2 = b1/b2 = c1/c2 | Coincident (same line) | Infinitely many | Consistent (dependent) |
Solving by elimination
Elimination method
- 1Multiply one or both equations so the coefficient of one variable becomes equal in size (matching or opposite signs).
- 2Add or subtract the equations to eliminate that variable.
- 3Solve the resulting single-variable equation.
- 4Substitute that value back into any original equation to find the other variable.
- 5Verify by putting both values into both equations.
📝 Solve 3x + 2y = 12 and 5x - 2y = 4
Step 1: The y-terms are +2y and -2y - already opposite, so add the two equations.
Step 2: (3x + 5x) + (2y - 2y) = 12 + 4, which gives 8x = 16.
Step 3: x = 16 / 8 = 2.
Step 4: Put x = 2 into 3x + 2y = 12: 6 + 2y = 12, so 2y = 6 and y = 3.
Step 5: Check in 5x - 2y = 4: 10 - 6 = 4. Correct. Solution: x = 2, y = 3.
Quick revision
- ✓A pair of linear equations = two straight lines; the solution is their meeting point.
- ✓Unique solution when a1/a2 is not equal to b1/b2 (lines intersect).
- ✓No solution when a1/a2 = b1/b2 but not equal to c1/c2 (parallel lines).
- ✓Infinitely many solutions when a1/a2 = b1/b2 = c1/c2 (same line).
- ✓Main algebraic methods: substitution, elimination, and cross-multiplication.
⚡ Quick check
For what value of k do the equations 2x + 3y = 5 and 4x + 6y = k have infinitely many solutions?
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