Two lines don’t always cross. This lesson joins substitution with what you learned about equations with no solution or infinitely many.
Why it works
Two lines in a plane can do only three things: cross once, never meet (parallel: same slope, different $y$-intercepts), or lie on top of each other (the same line). Algebra tells the same story: substitution ends with $x$ = a number, with a false statement, or with a true statement.
| The lines | Slopes and intercepts | Algebra ends with | Solutions |
|---|---|---|---|
| cross | different slopes | $x$ = a number | exactly one |
| are parallel | same slope, different $b$ | a false statement, like $-4 = 8$ | none |
| are the same line | same slope, same $b$ | a true statement, like $8 = 8$ | infinitely many |
Key idea
Write both equations as $y = mx + b$. Different slopes: one solution. Same slope and different intercepts: no solution. Same slope and same intercept: infinitely many solutions (every point on the line).
Worked examples
Worked example: Parallel lines
Solve: $y = 3x + 2$ and $6x - 2y = 8$.
- Rewrite the second: $-2y = -6x + 8$, so $y = 3x - 4$. Same slope 3, different intercepts: parallel.
- Algebra agrees: $6x - 2(3x + 2) = 8$ gives $-4 = 8$, which is false.
Answer: no solution
Worked example: The same line
Solve: $y = -x + 4$ and $2x + 2y = 8$.
- Rewrite the second: $2y = -2x + 8$, so $y = -x + 4$. It is the same line.
- Every point on it works, such as $(0, 4)$ and $(1, 3)$.
Answer: infinitely many solutions
Worked example: Tell at a glance
How many solutions do $y = 2x + 1$ and $y = -2x + 1$ have?
- The slopes, 2 and $-2$, are different, so the lines cross once (here at $(0, 1)$).
Answer: exactly one
Common mistake
Thinking “infinitely many solutions” means every pair $(x, y)$ works. Only points on the shared line do: $(0, 0)$ is not a solution of $y = -x + 4$.