Algebra
Systems of equations and inequalities
Classification of systems
20 practice questions
2 video lessons
Theory + worked examples
Theory
A two-line system falls into three types:
- One solution: the lines intersect (different slopes).
- No solution: the lines are parallel (same slope, different intercept).
- Infinitely many: the lines are identical (same equation).
Compare slopes and intercepts to classify without graphing.
Intersecting lines give one solution.
The three types of system.
Classifying by slope:
\[m_1\neq m_2:\text{one};\ \ m_1=m_2,b_1\neq b_2:\text{none};\ \ \text{same}:\infty\]
Consistent means it has a solution; inconsistent means none.
How to classify a system
- Write both equations in slope-intercept form.
- Compare the slopes.
- Different slopes: one solution.
- Same slope: check intercepts for none vs infinite.
Example 1 — One solution
How many solutions do \(y=x\) and \(y=-x+1\) have?
Solution
Different slopes cross once.
| \(\text{slopes } 1\neq-1\) | \(\Rightarrow\) | \(\text{one solution}\) |
Example 2 — No solution
How many solutions do \(y=x+1\) and \(y=x-3\) have?
Solution
Same slope, different intercept — parallel.
| \(\text{parallel}\) | \(\Rightarrow\) | \(\text{no solution}\) |
Example 3 — Infinitely many
How many solutions do \(y=2x+1\) and \(2y=4x+2\) have?
Solution
The second is the same line.
| \(\text{identical}\) | \(\Rightarrow\) | \(\text{infinite}\) |
Example 4 — Decide from slopes
How can slopes classify a system?
Solution
Different slopes: one solution. Same slope, different intercept: none. Same line: infinite.
Common pitfalls
Same slope, different intercept means no solution.
Identical equations mean infinitely many.
Compare in slope-intercept form to be sure.
Frequently asked questions
How many solutions can a linear system have?
One, none, or infinitely many.
When does a system have no solution?
When the lines are parallel.
When does it have infinitely many?
When the two equations are the same line.
How do you classify without graphing?
Compare slopes and intercepts.
More in Systems of equations and inequalities