Linear System Behavior Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Linear System Behavior.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The classification of a system of linear equations based on the geometric relationship of the lines: intersecting at one point (one unique solution), parallel with no intersection (no solution), or coincident/overlapping (infinitely many solutions).

Two lines can cross (one solution), be parallel (no solution), or overlap (infinite solutions).

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Linear system behavior reads a 2-line system's fate from how the lines sit: one solution, none, or infinitely many.

Common stuck point: The procedure for linear system behavior is the easy part; the trap is equating 'two equations' with 'one solution'. Asking "Am I classifying how many solutions a linear system has by how its lines relate?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I classifying how many solutions a linear system has by how its lines relate?

Worked Examples

Example 1

easy

Classify the system:

\begin{cases} 2x + y = 5 \\ 4x + 2y = 10 \end{cases}

Answer

Dependent (infinitely many solutions)

First step

Step 1: Check ratios:

\frac{2}{4} = \frac{1}{2} = \frac{5}{10}

Full solution

2
Step 2: All ratios equal: $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$ .
3
Step 3: The lines are identical — infinitely many solutions (dependent).

When all coefficient ratios are equal, the equations represent the same line. The system is consistent and dependent — every point on the line is a solution.

Example 2

medium

Classify:

\begin{cases} x - 3y = 2 \\ 2x - 6y = 7 \end{cases}

Example 3

medium

Classify the system:

\begin{cases} 3x + 2y = 8 \\ 6x + 4y = 16 \end{cases}

Example 4

medium

Classify:

\begin{cases} y = \tfrac{1}{2}x + 3 \\ x - 2y = -6 \end{cases}

Example 5

hard

For what

k

does

\begin{cases} 3x - 2y = 6 \\ 9x + ky = 18 \end{cases}

have NO unique solution?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Does

\begin{cases} x + y = 3 \\ x - y = 1 \end{cases}

have one solution, no solution, or infinitely many?

Example 2

medium

Classify:

\begin{cases} 3x - y = 4 \\ 6x - 2y = 8 \end{cases}

Example 3

easy

Two lines intersect at exactly one point. How many solutions does the system have?

Example 4

easy

Two parallel lines with different intercepts. How many solutions?

Example 5

easy

Two equations describe the exact same line. How many solutions?

Example 6

easy

Lines

y = 2x + 1

and

y = 3x - 4

: same slope or different?

Example 7

easy

Lines

y = 2x + 1

and

y = 2x + 5

: classify the system.

Example 8

easy

Lines

y = 2x + 1

and

2y = 4x + 2

: classify.

Example 9

easy

How many solutions can a system of two linear equations in two unknowns have?

Example 10

easy

Classify by graph: two lines drawn lie on top of each other. Solution count?

Example 11

medium

Classify:

\begin{cases} x + y = 4 \\ x - y = 2 \end{cases}

Example 12

medium

Classify:

\begin{cases} 2x + 3y = 6 \\ 4x + 6y = 7 \end{cases}

Example 13

medium

Classify:

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

Example 14

medium

Using slopes, predict the type of

\begin{cases} 3x - y = 1 \\ 6x - 2y = 5 \end{cases}

Example 15

medium

For what

k

does

\begin{cases} x + 2y = 3 \\ 2x + ky = 6 \end{cases}

have infinitely many solutions?

Example 16

medium

For what

k

does

\begin{cases} x + 2y = 3 \\ 2x + 4y = k \end{cases}

have NO solution?

Example 17

medium

A system reduces to

0 = 0

after elimination. What does this tell you?

Example 18

challenge

Three lines pairwise intersect but at three different points. Does the system have a common solution?

Example 19

challenge

Classify the

3\times3

system

\begin{cases} x+y+z=1 \\ 2x+2y+2z=2 \\ x - z = 0 \end{cases}

Example 20

challenge

Find all

a

for which

\begin{cases} x + y = 2 \\ ax + y = 4 \end{cases}

has a UNIQUE solution.

Example 21

medium

Classify

\begin{cases} y = 4x - 2 \\ y = 4x - 2 \end{cases}

Example 22

medium

Classify

\begin{cases} y = 5x + 1 \\ y = 5x + 1 \end{cases}

\begin{cases} y = 5x + 1 \\ y = 5x + 9 \end{cases}

Example 23

easy

Two lines have the same slope but different

y

-intercepts. How many solutions does the system have?

Example 24

easy

Two lines have different slopes. How many solutions does the system have?

Example 25

easy

Lines

y = -x + 3

and

y = 4x - 2

: classify the system.

Example 26

easy

Lines

y = 7x - 5

and

y = 7x - 5

: how many solutions?

Example 27

easy

Graphically, two lines coincide (lie on top of each other). The system is __________.

Example 28

easy

Lines

2x - y = 3

and

2x - y = 5

: classify.

Example 29

medium

Classify:

\begin{cases} 5x - 2y = 10 \\ 10x - 4y = 25 \end{cases}

Example 30

medium

For what value of

k

does

\begin{cases} 2x + 3y = 6 \\ 4x + 6y = k \end{cases}

have infinitely many solutions?

Example 31

medium

For what

k

does

\begin{cases} kx + 6y = 4 \\ 2x + 3y = 2 \end{cases}

have NO unique solution?

Example 32

medium

Classify:

\begin{cases} x - 2y = 4 \\ 3x + y = 5 \end{cases}

Example 33

medium

Lines

y = 3x + 2

and

y = 3x - 7

: how many points do they share?

Example 34

medium

For what

k

does

\begin{cases} 3x + ky = 12 \\ 6x + 8y = 24 \end{cases}

have infinitely many solutions?

Example 35

medium

A system has solution

(0, 0)

. Could the system also have other solutions? Under what condition?

Example 36

medium

Classify:

\begin{cases} 2x + 5y = 1 \\ 4x + 10y = 3 \end{cases}

Example 37

hard

For what values of

k

does

\begin{cases} x + ky = 1 \\ kx + y = 1 \end{cases}

have a UNIQUE solution?

Example 38

hard

For what

k

does

\begin{cases} x + y = 2 \\ x + ky = 2 \end{cases}

have infinitely many solutions?

Example 39

hard

3 \times 3

system has only the trivial solution

(0,0,0)

. What does that tell you about the coefficient matrix?

Example 40

hard

Three lines in the plane are pairwise non-parallel. Does the

3 \times 2

system necessarily have a common solution?

Example 41

hard

Find all

a

for which

\begin{cases} ax + 2y = 4 \\ 3x + (a+1)y = 6 \end{cases}

has infinitely many solutions.

Example 42

challenge

A system of two linear equations has solution set described by

(x, y) = (1 + 2t, 3 - t)

for all real

t

. What can you conclude about the system?

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Related Concepts

Systems of Equations Linear Functions Consistency Redundancy

Background Knowledge

These ideas may be useful before you work through the harder examples.

systems of equationslinear functions