Special Cases: No Solution & Infinite Solutions

TipLearning Objectives

By the end of this lesson, you’ll be able to:

  • Identify when a system has no solution or infinitely many solutions.
  • Recognize when two equations represent parallel lines or the same line.
  • Rewrite systems to compare slopes and intercepts efficiently.

Key Ideas

When solving systems, sometimes the algebra reveals that the system has no solution or infinitely many solutions. These special cases come directly from comparing the slopes and intercepts of the lines.

No Solution — Parallel Lines

Equations have the same slope but different intercepts:

\[ \begin{split} &y = mx + b_1 \\ &y = mx + b_2 \end{split} \]

These lines never meet → no solution.

Two parallel lines showing a system with no solution.

Infinite Solutions — Same Line

Equations represent the same line. One is just a multiple of the other.

Example:

\[ \begin{split} &2x + 4y = 8 \\ &x + 2y = 4 \end{split} \]

Second is the first divided by 2 → infinitely many solutions.

Common Problem Types

Parallel lines with different constants

Rewrite to slope-intercept and compare \(m\) and \(b\).

Equivalent equations

Check whether all coefficients and constants share the same ratio.

Systems that collapse during elimination

If you get a false statement (e.g., \(0 = 5\)), → no solution.
If you get a true identity (e.g., \(0 = 0\)), → infinite solutions.

Strategies

  • Always convert both equations to slope-intercept (\(y = mx + b\)) before comparing.
  • When in doubt, divide one equation by a constant to see if it matches the other.
  • During elimination:
    • False statements → no solution
    • True statements → infinitely many solutions

Worked Examples

Example 1 — No Solution

\[ \begin{cases} y = 5x - 1 \\ 10x - 2y = 8 \end{cases} \]

Rewrite second:

\[ -2y = -10x + 8 \Rightarrow y = 5x - 4 \]

Slopes match (5 and 5), intercepts differ → no solution.

Example 2 — Infinite Solutions

\[ \begin{cases} 3x + 6y = 12 \\ x + 2y = 4 \end{cases} \]

Divide the first equation by 3:

\[ x + 2y = 4 \]

Same exact equation → infinitely many solutions.

WarningCommon Mistakes
  • Assuming parallel lines eventually intersect “somewhere off the graph.”
  • Forgetting to compare both slope and intercept.
  • Not noticing when equations are scalar multiples.

Practice Problems

  1. Determine solution type:
    \(y = 4x + 1\), \(2y = 8x + 4\)

  2. Determine solution type:
    \(y = -3x + 5\), \(6x + 2y = 10\)

  3. Determine solution type:
    \(2x - 3y = 7\), \(4x - 6y = 14\)

1.
Rewrite second: \(y = 4x + 2\).
Same slope, different intercept → no solution.

2.
Rewrite second:
\(2y = 6x + 10 \Rightarrow y = 3x + 5\).
Slopes: \(-3\) vs \(3\) → different → one solution.

3.
Second is exactly twice the first → infinite solutions.

Summary

  • No solution when lines are parallel (same slope, different intercept).
  • Infinitely many solutions when equations represent the same line.
  • Often revealed by rewriting or by elimination leading to \(0=0\) (infinite) or \(0=5\) (none).
  • Compare slopes first—parallel lines appear immediately.
  • If every term in one equation is a scaled version of the other, the system has infinite solutions.
  • During elimination, watch for identity vs contradiction:
    • \(0=0\) → infinitely many
    • \(0=c\) (nonzero) → no solution