1Cademy - Verifying that $$3x - 2y = 6$$ and $$y = \frac{3}{2}x + 1$$ are Parallel

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Parallel Lines

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Verifying that $3x - 2y = 6$ and $y = \frac{3}{2}x + 1$ are Parallel

To determine whether the lines $3x - 2y = 6$ and $y = \frac{3}{2}x + 1$ are parallel, convert both equations to slope-intercept form and compare their slopes and $y$ -intercepts.

First equation: Solve $3x - 2y = 6$ for $y$ . Subtract $3x$ from both sides:

$-2y = -3x + 6$

Divide both sides by $-2$ :

$y = \frac{3}{2}x - 3$

The slope is $m = \frac{3}{2}$ and the $y$ -intercept is $(0, -3)$ .

Second equation: $y = \frac{3}{2}x + 1$ is already in slope-intercept form. The slope is $m = \frac{3}{2}$ and the $y$ -intercept is $(0, 1)$ .

Compare slopes and $y$ -intercepts:

First line: $m = \frac{3}{2}$ , $b = -3$
Second line: $m = \frac{3}{2}$ , $b = 1$

Because both lines share the same slope ( $\frac{3}{2}$ ) but have different $y$ -intercepts ( $-3$ vs. $1$ ), the lines are parallel. This example illustrates that when a standard-form equation has a coefficient other than $1$ or $-1$ on $y$ , dividing by that coefficient produces a fractional slope — and the parallelism check still works the same way: equal slopes with different $y$ -intercepts confirm the lines are parallel.

Updated 2026-05-03

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