Linear Equations in Two Variables: Methods & Solved Examples
A pair of linear equations in two variables is a topic that rewards practice more than memory. Once you know the three solving methods — substitution, elimination and graphing — and can tell at a glance whether a pair has one solution, none, or infinitely many, the whole chapter becomes routine marks in your Class 10 board exam. This guide explains each method step by step with worked examples and the rules for consistency.
What is a linear equation in two variables?
A linear equation in two variables is one that can be written in the form:
ax + by + c = 0 (where a and b are not both zero)
It involves two unknowns, usually x and y, each to the first power only. A single such equation has infinitely many solutions — every point on a straight line satisfies it. To pin down unique values for x and y you need a pair of equations considered together, which is why the chapter is about simultaneous equations.
Method 1: The substitution method
Here you make one variable the subject in one equation and substitute that expression into the other.
Example 1
Solve x + y = 7 and 2x − y = 2.
From the first equation, y = 7 − x. Substitute into the second: 2x − (7 − x) = 2, which gives 3x − 7 = 2, so 3x = 9 and x = 3. Then y = 7 − 3 = 4. The solution is x = 3, y = 4.
Method 2: The elimination method
Here you add or subtract the equations so that one variable cancels out. You may first multiply an equation by a number to make the coefficients match.
Example 2
Solve 3x + 2y = 12 and 5x − 2y = 4.
The y-coefficients are already +2 and −2, so adding the equations eliminates y: 8x = 16, giving x = 2. Substitute back into 3x + 2y = 12: 6 + 2y = 12, so 2y = 6 and y = 3. The solution is x = 2, y = 3.
Method 3: The graphical method
Each equation represents a straight line. To solve graphically, plot both lines on the same axes; the point where they intersect gives the solution.
- If the two lines intersect at one point, there is exactly one solution.
- If the lines are parallel, they never meet, so there is no solution.
- If the lines coincide (lie on top of each other), every point is shared, giving infinitely many solutions.
The graphical method is visual and intuitive but less precise when the solution is not a whole number, so the algebraic methods are usually preferred for accuracy.
Consistency: how many solutions are there?
Before solving, you can predict the number of solutions by comparing the coefficients of two equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0:
| Condition | Lines | Solutions | System |
|---|---|---|---|
| a₁/a₂ ≠ b₁/b₂ | Intersecting | Exactly one | Consistent |
| a₁/a₂ = b₁/b₂ ≠ c₁/c₂ | Parallel | None | Inconsistent |
| a₁/a₂ = b₁/b₂ = c₁/c₂ | Coincident | Infinitely many | Consistent (dependent) |
Example 3: Checking consistency
Are the equations 2x + 3y = 9 and 4x + 6y = 18 consistent?
Compare the ratios: a₁/a₂ = 2/4 = 1/2, b₁/b₂ = 3/6 = 1/2, and c₁/c₂ = 9/18 = 1/2. All three are equal, so the lines are coincident. The system is consistent with infinitely many solutions.
Choosing the right method
- Use substitution when one variable already has a coefficient of 1 and is easy to isolate.
- Use elimination when the coefficients line up for easy adding or subtracting.
- Use the graphical method when the question asks for a graph or when checking consistency visually.
Want to test yourself? Try the LearnIQ practice quiz for more mathematics questions, or browse all study guides.