Simultaneous equations with no solution
Use this page when the algebra seems to lead nowhere and you want to understand the case where two equations do not share a solution at all.
Start here if you want the short version before reading the full method.
- A simultaneous-equations system has no solution when there is no pair of values that satisfies both equations at once.
- On a graph, that usually means the two lines are parallel and never intersect.
What this topic means and what to look for first.
This case matters because it explains why some systems do not end with a clean x and y pair.
The graph view helps here because it shows why the algebra keeps resisting a shared answer: the lines never meet.
One reliable route through the topic.
- 1Solve the system as usual by elimination or substitution until you compare the two equations directly.
- 2Watch for a contradiction such as 0 = 5, which signals that the system is inconsistent.
- 3Interpret that contradiction as no shared intersection point.
- 4If helpful, connect the result to parallel lines on a graph.
See the method in action.
x + y = 4 and x + y = 7
- Subtract one equation from the other and you get 0 = 3.
- That contradiction shows there is no pair of values that can satisfy both equations.
- So the system has no solution.
y = 2x + 1 and y = 2x - 3
- Both lines have the same slope but different intercepts.
- Parallel lines never meet, so there is no intersection point.
- That means the system has no solution.
Things that commonly send the method off track.
- Thinking the system must always produce a pair of values.
- Treating a contradiction like 0 = 3 as an arithmetic mistake when it actually describes the system.
- Missing the graph meaning and not recognising the parallel-line case.
Use a short verification pass before moving on.
- If your algebra ends in a contradiction, check the line combination once more to confirm the arithmetic is right.
- If the contradiction holds, interpret it as no shared solution rather than forcing an x and y value.
Try a few variations before switching to a calculator or solver tool.
- x + y = 6 and x + y = 1
- y = 3x + 2 and y = 3x - 4
- 2x - y = 5 and 4x - 2y = 3
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Extra algebra revision resources
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Short answers worth checking.
It means there is no single pair of values that makes both equations true at the same time.
A contradiction like 0 = 3 or two parallel lines with different intercepts both show that the system has no solution.
Not always. Some systems genuinely have no shared solution, and the contradiction is the correct conclusion.
Continue with the next closely related topic.
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Check important answers independently before relying on them.