Q1. Solve 2x + 3y = 13 and x − y = −1.
From the second: x = y − 1
Substitute: 2(y − 1) + 3y = 13 ⇒ 5y = 15 ⇒ y = 3
x = 3 − 1 = 2. So (x, y) = (2, 3).
Pair of Linear Equations in Two Variables — Class 10
Two equations, two unknowns. Enter the coefficients and solve for x and y instantly — then learn every method and the consistency conditions.
Linear equation in two variablesAn equation of the form ax + by + c = 0; its graph is a straight line.
Methods of solvingGraphical, substitution, elimination and cross-multiplication all give the same solution (x, y).
ConsistencyCompare a₁/a₂, b₁/b₂, c₁/c₂: unique solution (a₁/a₂ ≠ b₁/b₂), infinitely many (all equal), or none (a₁/a₂ = b₁/b₂ ≠ c₁/c₂).
Two-equation solver Interactive
Enter a₁x + b₁y = c₁ and a₂x + b₂y = c₂ to solve for x and y.
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If a₁b₂ − a₂b₁ = 0 the lines are parallel or the same, so there is no unique solution.
Solved examples
Formula sheet
| Quantity | Formula |
|---|---|
| Cross-multiplication (x) | x = (b₁c₂ − b₂c₁)/(a₁b₂ − a₂b₁) |
| Cross-multiplication (y) | y = (c₁a₂ − c₂a₁)/(a₁b₂ − a₂b₁) |
| Unique solution if | a₁/a₂ ≠ b₁/b₂ |
Common mistakes & exam wins
- Substitution is easiest when one variable already has coefficient 1.
- For "unique / no / infinite solutions" questions, compare the ratios a₁/a₂, b₁/b₂, c₁/c₂.
- Parallel lines (a₁/a₂ = b₁/b₂ ≠ c₁/c₂) ⇒ inconsistent (no solution).
Frequently asked questions
How many solutions can a pair of linear equations have?
Exactly one (intersecting lines), infinitely many (coincident lines) or none (parallel lines).
When does a pair of linear equations have a unique solution?
When a₁/a₂ ≠ b₁/b₂, i.e. the lines intersect at one point.
What are the algebraic methods of solving?
Substitution, elimination and cross-multiplication.