Determine whether a system has one solution, no solution, or infinitely many solutions. | Systems of two linear equations in two variables | SAT Math

Use the equations or the line slopes to tell whether a system has one, none, or infinitely many solutions.

Core Idea

Compare the slopes: different slopes → one solution; same slope, different intercepts → no solution; same slope and same intercept → infinitely many solutions.

Understanding

When you try to solve a system and get a single pair $(𝑥, 𝑦)$ , the lines intersect once. But two other outcomes are possible.

If elimination or substitution produces a false statement like $0 = 5$ , the system has no solution. The lines are parallel — same slope, different intercepts — so they never meet.

If you get a true identity like $0 = 0$ , the system has infinitely many solutions. The two equations describe the same line, so every point on it works.

A fast shortcut: write both equations as $𝑎 𝑥 + 𝑏 𝑦 = 𝑐$ . If $𝑎 1 𝑎 2 = 𝑏 1 𝑏 2 \neq 𝑐 1 𝑐 2$ , no solution. If all three ratios are equal, infinitely many. Otherwise, exactly one solution.

Step by Step

Attempt elimination or substitution as usual.
If you reach a statement like $0 = 5$ (false), the answer is no solution.
If you reach $0 = 0$ (always true), the answer is infinitely many solutions.
If you find a specific value for one variable, there is exactly one solution — continue solving.

Misconceptions

Seeing $0 = 0$ and thinking you made an error — it actually means infinitely many solutions, not a dead end.
Concluding 'no solution' just because the algebra feels complicated; no solution only occurs when you reach a contradiction.
Assuming every system must have a solution — parallel lines with different intercepts never intersect.

Question

Worked Example

How many solutions does the system $4 𝑥 - 6 𝑦 = 10$ and $- 2 𝑥 + 3 𝑦 = 8$ have?

Select an answer to see the explanation