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Determine whether lines are parallel using slope

Parallel lines in the coordinate plane have the same slope and different intercepts when written in slope-intercept form, so they never intersect; two distinct vertical lines are also parallel because both have undefined slope. This reasoning connects graphical, tabular, and equation representations and distinguishes parallelism from coincident lines or perpendicular lines, supporting the study of linear systems and coordinate geometry.

Detailed Explanation: Determine whether lines are parallel using slope

To determine whether two lines are parallel, write both equations in slope-intercept form:

y=mx+by=mx+b

Here, mm is the slope and bb is the yy-intercept.

Example: Are the lines

y=2x+3y=2x+3

and

2x−y=−12x-y=-1

parallel?

Step 1: Identify the slope of the first line.

The first equation is already in slope-intercept form:

y=2x+3y=2x+3

So its slope is

m1=2m_1=2

and its yy-intercept is 33.

Step 2: Rewrite the second equation in slope-intercept form.

Start with

2x−y=−12x-y=-1

Subtract 2x2x from both sides:

−y=−2x−1-y=-2x-1

Multiply by −1-1:

y=2x+1y=2x+1

So the second line has slope

m2=2m_2=2

and yy-intercept 11.

Step 3: Compare the slopes and intercepts.

The slopes are equal:

m1=m2=2m_1=m_2=2

The yy-intercepts are different: 3≠13\ne1.

Therefore, the lines have the same slope but different intercepts, so they are distinct parallel lines.

The lines are parallel.\boxed{\text{The lines are parallel.}}

Lines with the same slope and the same intercept are actually the same line, not two different parallel lines. Vertical lines are also parallel when they are distinct, because both have undefined slope.

Learn by doing: Determine whether lines are parallel using slope

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