Some of One, None of the Other

Learning Focus

Recognize systems with no solution or infinitely many solutions by comparing slopes and intercepts.

Big idea

Two lines in a plane can intersect once, never, or everywhere.

• One solution: different slopes.
• No solution: same slope, different $y$-intercept (parallel). Algebra gives a false statement such as $0=8$.
• Infinitely many: same slope and same intercept (same line). Algebra gives a true statement such as $0=0$.

Worked example

Solve $y=2x+3$ and $y=2x-5$.

Set equal: $2x+3=2x-5 \Rightarrow 3=-5$, which is false. The lines are parallel, so there is no solution.

Now $2x+4y=8$ and $x+2y=4$: the first is $2$ times the second, so they are the same line and have infinitely many solutions.

Takeaways

• Same slope, different intercept $\Rightarrow$ no solution.
• Same slope and intercept $\Rightarrow$ infinitely many.
• A false statement means none; a true identity means infinitely many.