Section 5.5: Special Cases in Systems
Some systems of linear equations do not have a single solution. These are called special cases:
- No Solution: Lines are parallel and never intersect.
- Infinite Solutions: Lines coincide (are the same line).
Example 1: No Solution
Solve:
\( 2x + 3y = 6 \)
\( 4x + 6y = 10 \)
Step 1: Multiply first equation by 2: \( 4x + 6y = 12 \)
Step 2: Compare with second equation \( 4x + 6y = 10 \)
Step 3: Contradiction → no solution. The lines are parallel.
Example 2: Infinite Solutions
Solve:
\( x - y = 2 \)
\( 2x - 2y = 4 \)
Step 1: Divide second equation by 2 → \( x - y = 2 \)
Step 2: Both equations are identical → infinitely many solutions along the line \( x - y = 2 \).
Practice Problems
- Determine if the system has one solution, no solution, or infinite solutions: \( 3x + 2y = 5 \), \( 6x + 4y = 10 \).
- Determine for \( x + y = 4 \), \( 2x + 2y = 8 \).
- Determine for \( x - 2y = 1 \), \( 2x - 4y = 5 \).
- Determine for \( 5x + 3y = 7 \), \( 10x + 6y = 14 \).
- Determine for \( 2x - y = 3 \), \( 4x - 2y = 6 \).