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

How the solutions of a linear system relate to the geometric arrangement of the lines.

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: The three cases: consistent-independent, inconsistent, consistent-dependent.

Common stuck point: Parallel lines mean same slope, different intercept \to no solution.

Sense of Study hint: Compare the slopes of the two lines first. Same slope means either no solution or infinitely many.

Worked Examples

Example 1

easy
Classify the system: \begin{cases} 2x + y = 5 \\ 4x + 2y = 10 \end{cases}

Solution

  1. 1
    Step 1: Check ratios: \frac{2}{4} = \frac{1}{2} = \frac{5}{10}.
  2. 2
    Step 2: All ratios equal: \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}.
  3. 3
    Step 3: The lines are identical — infinitely many solutions (dependent).

Answer

Dependent (infinitely many solutions)
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}

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}
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Background Knowledge

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

systems of equationslinear functions