How Many Solutions Does a System Have?
A system of two linear equations can end three ways: exactly one solution, no solution, or infinitely many solutions. There is no fourth option — two straight lines can't cross exactly twice, or exactly five times.
The best part is that you can tell which case you have without solving anything. The number of solutions is just the number of points where the two lines meet, and slopes and -intercepts tell you that instantly.
The three cases
Different slopes: the lines are tilted differently, so they must cross exactly once. The system has exactly one solution.
Same slope, different -intercepts: the lines are parallel — same tilt, different starting heights — so they never meet. The system has no solution. The graph below shows this case: and run side by side forever.
Same slope and same -intercept: the two equations draw the exact same line. Every point on that line works in both equations, so the system has infinitely many solutions.
Compare slopes first, then intercepts
Put both equations in slope-intercept form and compare the values. If the slopes are different, stop — the answer is exactly one solution, and the intercepts don't matter.
If the slopes match, compare the values. Different -intercepts mean parallel lines and no solution. Matching -intercepts mean it's the same line and infinitely many solutions.
Watch for the same line in disguise
An equation like hides its slope. Divide both sides by and it becomes — the same line as , even though the two equations looked different at first glance.
So before you compare anything, rewrite each equation so is alone on the left. One quick division or subtraction is all it usually takes, and it keeps you from calling two copies of the same line "parallel."
Worked examples
Example 1: different slopes
How many solutions does the system and have?
Answer: Exactly one solution.
Example 2: same slope, different intercepts
How many solutions does the system and have?
Answer: No solution.
Example 3: the same line in disguise
How many solutions does the system and have?
Answer: Infinitely many solutions.
Try one yourself
Common questions
Can a system of two lines have exactly two solutions?
No. Two distinct straight lines can cross at most once. If you ever find two different points that work in both equations, the "two" lines are actually the same line — and then every point on it works, giving infinitely many solutions.
Does infinitely many solutions mean any pair of numbers works?
No. Only the points on the shared line work. and have infinitely many solutions, but still fails both — it isn't on the line. Infinitely many candidates, not all candidates.
What if the equations aren't in form?
Rewrite them first. Comparing slopes only works once each equation is solved for . Move the -term across, divide by the coefficient of , and then compare and .
What does no solution look like when you solve algebraically?
The variables cancel and you're left with a false statement like . That contradiction is the algebra's way of saying the lines are parallel — no pair can make both equations true.
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