Systems of Equations: No Solution Systems
SAT Key Concept: A system of equations has no solution when the equations represent parallel lines — lines that never intersect. This topic appears frequently on the SAT, often disguised in word problems or with variables in coefficients.
What Does "No Solution" Mean?
A system of two linear equations can have:
| Type | Solutions | Lines |
|---|---|---|
| One Solution | Exactly 1 | Intersect at one point |
| No Solution | 0 | Parallel (never meet) |
| Infinite Solutions | ∞ | Same line (overlap) |
When a system has no solution, we say the system is inconsistent.
The Golden Rule
For the system:
$$\begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases}$$
There is no solution when:
$$\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$$
In plain English: The coefficients of $x$ and $y$ are proportional (same slope), but the constants are NOT proportional (different y-intercepts).
In slope-intercept form: Same slope, different y-intercepts → parallel lines.
Visualizing Parallel Lines (No Solution)
The graph below shows two parallel lines. Notice they have the same slope but different y-intercepts, so they never cross — meaning there is no $(x, y)$ pair that satisfies both equations simultaneously.
Interactive visualization — available in the full lesson