Linear System Behavior Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

easy
Classify the system: {2x+y=54x+2y=10\begin{cases} 2x + y = 5 \\ 4x + 2y = 10 \end{cases}

Solution

  1. 1
    Step 1: Check ratios: 24=12=510\frac{2}{4} = \frac{1}{2} = \frac{5}{10}.
  2. 2
    Step 2: All ratios equal: a1a2=b1b2=c1c2\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.

About Linear System Behavior

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).

Learn more about Linear System Behavior →

More Linear System Behavior Examples

Example 2 medium

Classify: [formula]

Example 3 easy

Does [formula] have one solution, no solution, or infinitely many?

Example 4 medium

Classify: [formula]

← All Linear System Behavior Examples Linear System Behavior Concept