Systems with No Solution or Infinitely Many Solutions

Three Possibilities for Linear Systems

Type	Lines	Solutions	Algebraic Result
Consistent & independent	Intersect at one point	Exactly one	$x = a$, $y = b$
Inconsistent	Parallel (never meet)	None	$0 = k$ (false statement)
Consistent & dependent	Same line (coincident)	Infinitely many	$0 = 0$ (true statement)

Parallel Lines — No Solution

Two lines are parallel when they have the same slope but different y-intercepts.

$y = 3x + 2 \quad \text{and} \quad y = 3x + 7$

Both have slope 3 but different y-intercepts. They never intersect → no solution.

In standard form: $Ax + By = C_1$ and $Ax + By = C_2$ (where $C_1 \neq C_2$) → no solution.

Same Line — Infinitely Many Solutions

Two equations represent the same line when one is a multiple of the other.

$y = 3x + 2 \quad \text{and} \quad 2y = 6x + 4$

The second equation is just 2× the first. Every point on the line satisfies both equations → infinitely many solutions.

Using the Ratio Test

For the system $a_1x + b_1y = c_1$ and $a_2x + b_2y = c_2$:

• If $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$: one solution
• If $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$: no solution (parallel)
• If $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$: infinitely many solutions (same line)

Finding a Parameter for No Solution

Example: For what value of $k$ does the system have no solution?

$2x + 3y = 7$ $kx + 6y = 10$

For no solution, the lines must be parallel: same slope, different y-intercepts.

Slope of first: $-\frac{2}{3}$

Slope of second: $-\frac{k}{6}$

For parallel: $-\frac{k}{6} = -\frac{2}{3}$ → $k = 4$

Check constants: $\frac{7}{10} \neq \frac{3}{6}$, so $k = 4$ gives no solution ✓

Finding a Parameter for Infinite Solutions

Example: For what values of $a$ and $b$ does the system have infinitely many solutions?

$3x + 6y = 15$ $ax + by = 5$

For infinite solutions: $\frac{3}{a} = \frac{6}{b} = \frac{15}{5}$

$\frac{15}{5} = 3$, so $\frac{3}{a} = 3$ → $a = 1$ and $\frac{6}{b} = 3$ → $b = 2$.

Tip 1: Convert to Slope-Intercept Form

The quickest way to spot parallel or identical lines is to compare slopes and y-intercepts.

Tip 2: Use the Ratio Test for Standard Form

When both equations are in standard form, compare the ratios of corresponding coefficients.

Tip 3: If You're Solving and Variables Cancel

If you're working through substitution or elimination and the variable terms cancel:

• False statement remaining → no solution
• True statement remaining → infinitely many solutions

Tip 4: The SAT Loves Parameter Questions

Expect questions like "For what value of $k$..." Be ready to set up slope equality conditions.

Tip 5: Graphically Think About It

Parallel = same direction, never meet. Same line = perfect overlap. One solution = they cross once.