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Example

Verifying that y=2x3y = 2x - 3 and 6x+3y=9-6x + 3y = -9 are the Same Line

To determine whether the lines y=2x3y = 2x - 3 and 6x+3y=9-6x + 3y = -9 are parallel, convert both equations to slope-intercept form and compare their slopes and yy-intercepts.

First equation: y=2x3y = 2x - 3 is already in slope-intercept form. The slope is m=2m = 2 and the yy-intercept is (0,3)(0, -3).

Second equation: Solve 6x+3y=9-6x + 3y = -9 for yy. Add 6x6x to both sides:

3y=6x93y = 6x - 9

Divide every term by 33:

y=2x3y = 2x - 3

The slope is m=2m = 2 and the yy-intercept is (0,3)(0, -3).

Compare slopes and yy-intercepts:

  • First line: m=2m = 2, yy-intercept (0,3)(0, -3)
  • Second line: m=2m = 2, yy-intercept (0,3)(0, -3)

Both lines share the same slope and the same yy-intercept, so the two equations represent the same line — they are coincident. The lines are not parallel; they are identical.

This example demonstrates that when the parallelism check reveals matching slopes and matching yy-intercepts, the equations do not describe two separate parallel lines but rather a single line expressed in two different algebraic forms.

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Updated 2026-05-03

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