Linear System Behavior Math Example 3

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

Example 3

easy
Does {x+y=3xโˆ’y=1\begin{cases} x + y = 3 \\ x - y = 1 \end{cases} have one solution, no solution, or infinitely many?

Solution

  1. 1
    11โ‰ 1โˆ’1\frac{1}{1} \neq \frac{1}{-1}. The ratios differ, so the lines intersect.
  2. 2
    One unique solution: adding gives 2x=42x = 4, x=2x = 2, y=1y = 1.

Answer

One solution: (2,1)(2, 1)
When the coefficient ratios are not equal (a1a2โ‰ b1b2\frac{a_1}{a_2} \neq \frac{b_1}{b_2}), the lines are not parallel and intersect at exactly one point.

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 2 medium

Classify: [formula]

Example 4 medium

Classify: [formula]

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