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Verifying that and are the Same Line
To determine whether the lines and are parallel, convert both equations to slope-intercept form and compare their slopes and -intercepts.
First equation: is already in slope-intercept form. The slope is and the -intercept is .
Second equation: Solve for . Add to both sides:
Divide every term by :
The slope is and the -intercept is .
Compare slopes and -intercepts:
- First line: , -intercept
- Second line: , -intercept
Both lines share the same slope and the same -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 -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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