Systems of Equations with No Solution / Infinite Solutions on SAT
Master the coefficient ratio shortcut for linear systems.
For a system of linear equations (a₁x + b₁y = c₁ and a₂x + b₂y = c₂): 1. No Solution (Parallel Lines): Slopes are equal and constants differ (a₁/a₂ = b₁/b₂ ≠ c₁/c₂). 2. Infinitely Many Solutions (Same Line): Equations are identical multiples (a₁/a₂ = b₁/b₂ = c₁/c₂). 3. One Solution (Intersecting Lines): Slopes differ (a₁/a₂ ≠ b₁/b₂).
These questions frequently ask you to find the value of a constant "k" that makes a system have no solution. Setting the coefficient ratio equal solves them in 10 seconds.
Reference Data Table
| Condition | Geometric Meaning | Coefficient Ratio Rule |
|---|---|---|
| No Solution | Parallel Lines (Never Intersect) | a₁ / a₂ = b₁ / b₂ ≠ c₁ / c₂ |
| Infinitely Many Solutions | Coincident Lines (Same Line) | a₁ / a₂ = b₁ / b₂ = c₁ / c₂ |
| Exactly One Solution | Intersecting Lines (One Point) | a₁ / a₂ ≠ b₁ / b₂ |
Key Takeaways & Pro Tips
Frequently asked questions
How do you check no solution in Desmos?
Graph both equations in Desmos. If the lines are parallel and never cross, there is no solution.
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