In the equation above, [latex]m=1[/latex] and [latex]b=−5[/latex].

Since [latex]m=1[/latex], the slope is [latex]1[/latex].

To find the slope of a parallel line, use the same slope.

The slope of the parallel line is [latex]1[/latex].

Use the method for writing an equation from the slope and a point on the line. Substitute 1 for m, and the point [latex](−2,1)[/latex] for [latex]x[/latex] and [latex]y[/latex].

Write the equation using the new slope for m and the b you just found.

Answer

[latex]y=x+3[/latex]

Determine the Equation of a Line Parallel to Another Line Through a Given Point

Determine the Equation of a Line Perpendicular to Another Line Through a Given Point

When you are working with perpendicular lines, you will usually be given one of the lines and an additional point. Remember that two non-vertical lines are perpendicular if the slope of one is the negative reciprocal of the slope of the other. To find the slope of a perpendicular line, find the reciprocal, and then find the opposite of this reciprocal. In other words, flip it and change the sign.

Example

Write the equation of a line that contains the point [latex](1,5)[/latex] and is perpendicular to the line [latex]y=2x– 6[/latex].

Show Solution

Identify the slope of the line you want to be perpendicular to.

The given line is written in [latex]y=mx+b[/latex] form, with [latex]m=2[/latex] and [latex]b=-6[/latex]. The slope is [latex]2[/latex].

To find the slope of a perpendicular line, find the reciprocal, [latex] \displaystyle \frac{1}{2}[/latex], then the opposite, [latex] \displaystyle -\frac{1}{2}[/latex].

The slope of the perpendicular line is [latex] \displaystyle -\frac{1}{2}[/latex].

Use the method for writing an equation from the slope and a point on the line. Substitute [latex] \displaystyle -\frac{1}{2}[/latex] for m, and the point [latex](1,5)[/latex] for [latex]x[/latex] and [latex]y[/latex].

Write the equation using the new slope for m and the b you just found.

Answer

[latex]y=-\frac{1}{2}x+\frac{11}{2}[/latex]

Determine the Equation of a Line Perpendicular to a Line in Slope-Intercept Form

Example

Write the equation of a line that is parallel to the line [latex]y=4[/latex] through the point [latex](0,10)[/latex].

Show Solution

Rewrite the line into [latex]y=mx+b[/latex] form, if needed.

You may notice without doing this that [latex]y=4[/latex] is a horizontal line 4 units above the x-axis. Because it is horizontal, you know its slope is zero.

In the equation above, [latex]m=0[/latex] and [latex]b=4[/latex].

Since [latex]m=0[/latex], the slope is [latex]0[/latex]. This is a horizontal line.

To find the slope of a parallel line, use the same slope.

The slope of the parallel line is also [latex]0[/latex].

Since the parallel line will be a horizontal line, its form is

[latex]y=\text{a constant}[/latex]

Since we want this new line to pass through the point [latex](0,10)[/latex], we will need to write the equation of the new line as:

[latex]y=10[/latex]

This line is parallel to [latex]y=4[/latex] and passes through [latex](0,10)[/latex].

Answer

[latex]y=10[/latex]

Example

Write the equation of a line that is perpendicular to the line [latex]y=-3[/latex] through the point [latex](-2,5)[/latex].

Show Solution

In the equation above, [latex]m=0[/latex] and [latex]b=-3[/latex].

A perpendicular line will have a slope that is the negative reciprocal of the slope of [latex]y=-3[/latex], but what does that mean in this case?

The reciprocal of [latex]0[/latex] is [latex]\frac{1}{0}[/latex], but we know that dividing by [latex]0[/latex] is undefined.

This means that we are looking for a line whose slope is undefined, and we also know that vertical lines have slopes that are undefined. This makes sense since we started with a horizontal line.

The form of a vertical line is [latex]x=\text{a constant}[/latex], where every x-value on the line is equal to some constant. Since we are looking for a line that goes through the point [latex](-2,5)[/latex], all of the [latex]x[/latex]-values on this line must be [latex]-2[/latex].

The equation of a line passing through [latex](-2,5)[/latex] that is perpendicular to the horizontal line [latex]y=-3[/latex] is therefore,

[latex]x=-2[/latex]

Answer

[latex]x=-2[/latex]

Try It

Find the Equation of a Perpendicular and Horizontal Line to a Horizontal Line

Summary

When lines in a plane are parallel (that is, they never cross), they have the same slope. When lines are perpendicular (that is, they cross at a [latex]90°[/latex] angle), their slopes are opposite reciprocals of each other. The product of their slopes will be [latex]-1[/latex], except in the case where one of the lines is vertical causing its slope to be undefined. You can use these relationships to find an equation of a line that goes through a particular point and is parallel or perpendicular to another line.

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