Linear System Behavior Math Example 2

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

Example 2

medium
Classify: {xโˆ’3y=22xโˆ’6y=7\begin{cases} x - 3y = 2 \\ 2x - 6y = 7 \end{cases}

Solution

  1. 1
    Step 1: 12=โˆ’3โˆ’6=12\frac{1}{2} = \frac{-3}{-6} = \frac{1}{2} but 27โ‰ 12\frac{2}{7} \neq \frac{1}{2}.
  2. 2
    Step 2: Coefficient ratios equal but constant ratio different: parallel lines.
  3. 3
    Step 3: No intersection โ€” inconsistent (no solution).

Answer

Inconsistent (no solution)
When a1a2=b1b2โ‰ c1c2\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}, the lines are parallel but distinct โ€” they never meet. The system has no 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 โ†’

More Linear System Behavior Examples

Example 1 easy

Classify the system: [formula]

Example 3 easy

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

Example 4 medium

Classify: [formula]

โ† All Linear System Behavior Examples Linear System Behavior Concept