Most systems of two linear equations have exactly one solution. Two special cases don’t: parallel lines, which never meet, and two equations that describe the same line.
No solution: parallel lines
Lines with the same slope but different y-intercepts are parallel. They never cross, so no point lies on both.
Example 1: by substitution
Solve y = 2x + 1 and y = 2x − 3.
Set them equal: 2x + 1 = 2x − 3. Subtract 2x from both sides: 1 = −3.
That is impossible, so there is no solution.
Example 2: by elimination
Solve 2x + 4y = 8 and x + 2y = 3.
Multiply the second equation by 2: 2x + 4y = 6. Subtract it from the first: 0 = 2.
That is false, so there is no solution. Both lines have slope −1/2.
See parallel lines in the Intersection Finder →
Infinitely many solutions: the same line
Sometimes two equations look different but describe exactly the same line. Every point on that line satisfies both equations, so there are infinitely many solutions.
Example 3
Solve 3x − y = 2 and 6x − 2y = 4.
The second equation is the first one multiplied by 2. Multiply the first by 2 and subtract: 0 = 0.
That is always true, so there are infinitely many solutions: every point on the line y = 3x − 2.
See two equations for the same line →
How to tell before you start
Write both equations in the form y = mx + b and compare:
- Different slopes: exactly one solution.
- Same slope, different y-intercepts: no solution.
- Same slope and same y-intercept: infinitely many solutions.
For equations in standard form, a₁x + b₁y = c₁ and a₂x + b₂y = c₂, compare the ratios. If a₁/a₂ = b₁/b₂ but that ratio is different from c₁/c₂, there is no solution. If all three ratios are equal, there are infinitely many solutions.
What the algebra tells you
In both special cases, the variables cancel out completely while you are solving. Look at what is left:
- A false statement, such as 1 = −3 or 0 = 2, means no solution.
- A true statement, such as 0 = 0 or 4 = 4, means infinitely many solutions.
Teachers sometimes call these systems inconsistent (no solution) and dependent (infinitely many). A system with exactly one solution is consistent and independent.