4.5Special Cases of Systems
Identify and solve systems with no solution, one solution, or infinitely many solutions — graphically and algebraically.
One Solution — lines intersect at (1, 2)
No Solution — parallel lines never meet
Infinitely Many Solutions — same line
Three possible outcomes for any system
Why This Matters
Not every system of equations has exactly one answer. Some systems have no solution at all — the lines are parallel and never cross. Others have infinitely many solutions — the two equations describe the same line. Recognizing these special cases quickly saves time on tests and prevents you from chasing an answer that doesn't exist.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
How can we identify whether a system of equations has one solution, no solution, or infinitely many solutions — both graphically and algebraically?
Lesson Overview
A system of two linear equations can have exactly one solution, no solution, or infinitely many solutions. These outcomes depend on whether the lines intersect, are parallel, or are the same line. Recognizing these special cases quickly — both graphically and algebraically — saves time on tests and prevents you from chasing an answer that doesn't exist.
Three possible outcomes for any system
Identify special cases algebraically
No Solution — parallel lines never intersect (same slope, different y-intercepts)
Infinitely Many Solutions — same line (equations are equivalent)
Identifying Graphically
- 1 intersection point → One solution
- 0 intersections (parallel) → No solution
- Lines overlap → Infinitely many
Identifying Algebraically
- Unique x and y → One solution
- False statement (e.g. 0 = 5) → No solution
- True statement (e.g. 0 = 0) → Infinitely many
Worked Examples
Solve the system and classify it: y = 3x + 2 and y = 3x − 5
Set the equations equal: 3x + 2 = 3x − 5
Subtract 3x from both sides: 2 = −5
This is a false statement — the lines are parallel.
Solve the system and classify it: 2x + y = 4 and 4x + 2y = 8
Multiply the first equation by 2: 4x + 2y = 8
Subtract from the second equation: 4x + 2y − (4x + 2y) = 8 − 8
0 = 0 — this is always true.
Solve the system and classify it: x + y = 5 and x − y = 1
Add the equations: 2x = 6, so x = 3
Substitute x = 3 into x + y = 5: 3 + y = 5, so y = 2
Solution: (3, 2). The lines intersect at one point.
Without solving, classify: y = −2x + 7 and 2y = −4x + 14
Rewrite the second equation: divide both sides by 2 → y = −2x + 7
Both equations are identical.
Classify by comparing slopes and intercepts: 3x − y = 6 and 6x − 2y = 10
Rewrite first: y = 3x − 6 (slope 3, y-intercept −6)
Rewrite second: 2y = 6x − 10 → y = 3x − 5 (slope 3, y-intercept −5)
Same slope, different y-intercepts → parallel lines.
Guided Practice
Guided Practice Video: Special Cases of Systems
Watch the guided practice walkthrough for special cases of systems, then complete the problems below.
Video by Sang Real Math
Watch on YouTube ↗Classify the system without fully solving: y = 4x − 1 and y = 4x + 3
Hint: Compare the slopes and y-intercepts. Same slope but different y-intercepts means the lines are parallel.
Solve and classify: x + 2y = 6 and 2x + 4y = 12
Hint: Try multiplying the first equation by 2. If the result matches the second equation exactly, the lines are the same.
Solve and classify: 3x − y = 4 and x + y = 8
Hint: Add the equations to eliminate y. If you get a unique value for x, the system has one solution.
Classify without solving: 5x − 2y = 10 and 10x − 4y = 25
Hint: Multiply the first equation by 2. Compare the result to the second equation — are the constants the same?
Solve and classify: 4x + y = 7 and 8x + 2y = 14
Hint: Multiply the first equation by 2. What do you notice when you compare it to the second equation?
Key Vocabulary
One Solution
The lines intersect at exactly one point. The system is called consistent and independent.
No Solution
The lines are parallel — same slope, different y-intercepts. The system is called inconsistent.
Infinitely Many Solutions
The lines are the same — same slope and same y-intercept (or equivalent equations). The system is called consistent and dependent.
Parallel Lines
Two lines with the same slope but different y-intercepts. They never intersect.
Coincident Lines
Two lines that are identical — every point on one line is also on the other.
Interactive Practice — 5 Questions
A system of equations has no solution when the lines are:
When solving a system algebraically and you get 0 = 0, the system has:
Classify: y = 7x − 2 and y = 7x + 5
Classify: 3x + y = 9 and 6x + 2y = 18
Which system has exactly one solution?
Independent Practice
Independent Practice
Classify each system without fully solving: (a) y = 2x + 1 and y = 2x − 3; (b) y = 5x − 4 and 2y = 10x − 8
Solve and classify: 4x + y = 7 and 8x + 2y = 14
Solve and classify: x − 3y = 6 and 2x − 6y = 9
Solve and classify: 2x + 5y = 10 and x + 2y = 4
Write a system of two equations that has no solution. Explain how you know.
Common Mistakes
Stopping when you get 0 = 0 and saying "no solution"
0 = 0 is always TRUE — that means infinitely many solutions, not no solution.
Stopping when you get 0 = 5 and saying "infinitely many"
0 = 5 is always FALSE — that means no solution.
Assuming parallel lines because the equations look similar
Always rewrite both equations in slope-intercept form and compare slopes AND y-intercepts before classifying.
Math Tips
Rewrite both equations in slope-intercept form (y = mx + b) before comparing — this makes classification fast.
Same slope, different y-intercepts → parallel lines → no solution (inconsistent system).
Same slope AND same y-intercept (or equivalent equations) → same line → infinitely many solutions (dependent system).
When you get 0 = 0 algebraically, it means infinitely many solutions — not no solution.
When you get a false statement like 0 = 5 algebraically, it means no solution — the lines are parallel.