Horizontal and Vertical Lines (Examples)

A horizontal line goes straight across, and its equation is $y = b$ , where b is the y-value of every point on it. A vertical line goes straight up and down, and its equation is $x = a$ . Horizontal lines have a slope of 0. Vertical lines have an undefined slope.

Before this lesson: You should know how to plot points on the coordinate plane and how to find slope from two points.

Extra practice: The Algebra 1 Unit 4 Practice Pack (Part 1) has printable problems, answer keys and a cheat sheet for this unit.

What Are Horizontal and Vertical Lines?

A horizontal line is flat, like the horizon. Look at these points: (−3, 2), (0, 2) and (4, 2). The x-values change, but every y-value is 2. So the line through them is $y = 2$ .

A vertical line stands straight up, like a flagpole. Look at these points: (1, −2), (1, 0) and (1, 5). The y-values change, but every x-value is 1. So the line through them is $x = 1$ .

The horizontal line y = 3 and the vertical line x = −2 on a coordinate grid. They cross at the point (−2, 3). — The line y = 3 goes across. The line x = −2 goes up and down. They cross at (−2, 3).

Here is a simple way to remember it. The equation tells you which value stays the same.

$y = 3$ means “y is always 3.” The x-values can be anything, so the line goes sideways.
$x = -2$ means “x is always −2.” The y-values can be anything, so the line goes up and down.

Equations of Horizontal and Vertical Lines

Line	Equation	Slope	Where it crosses an axis
Horizontal	$y = b$	0	Crosses the y-axis at (0, b)
Vertical	$x = a$	Undefined	Crosses the x-axis at (a, 0)

Two special lines are worth knowing:

The x-axis is the horizontal line $y = 0$ .
The y-axis is the vertical line $x = 0$ .

Why Their Slopes Are 0 and Undefined

Use the slope formula on two points from each line.

For the horizontal line through (−3, 2) and (4, 2):

m = \frac{2 - 2}{4 - (-3)} = \frac{0}{7} = 0

The rise is 0, so the slope is 0.

For the vertical line through (1, −2) and (1, 5):

m = \frac{5 - (-2)}{1 - 1} = \frac{7}{0}

The run is 0, and you can’t divide by 0. So the slope of a vertical line is undefined.

Worked Examples

Example 1: Horizontal line through a point

Write the equation of the horizontal line through (4, −5).

Step 1. On a horizontal line, every point has the same y-value.

Step 2. Read the y-value of the point (4, −5). It is −5.

y = -5

Step 3. Write the equation. Every point on the line has y equal to −5.

Answer: $y = -5$

Example 2: Vertical line through a point

Write the equation of the vertical line through (−3, 6).

Step 1. On a vertical line, every point has the same x-value.

Step 2. Read the x-value of the point (−3, 6). It is −3.

x = -3

Step 3. Write the equation. Every point on the line has x equal to −3.

Answer: $x = -3$

Example 3: Line through two points

Write the equation of the line through (2, 7) and (2, −1). What is its slope?

Step 1. Compare the points. Both x-values are 2, but the y-values are different.

x = 2

Step 2. When the x-values match, the line is vertical. Write the equation with that x-value.

m = \frac{-1 - 7}{2 - 2}

Step 3. Find the slope with the slope formula.

m = \frac{-8}{0}

Step 4. Subtract on the top and on the bottom.

Step 5. The bottom is 0, and you can’t divide by 0.

Answer: $x = 2$ . The slope is undefined.

Example 4: Where two lines cross

Graph $y = -2$ and $x = 4$ . Where do they cross?

The horizontal line y = −2 and the vertical line x = 4 cross at the point (4, −2). — The lines y = −2 and x = 4 cross at the point (4, −2).

Step 1. Draw $y = -2$ as a flat line through −2 on the y-axis.

Step 2. Draw $x = 4$ as an up-and-down line through 4 on the x-axis.

Step 3. Take the x-value from $x = 4$ . Every point on that line has x = 4.

Step 4. Take the y-value from $y = -2$ . Every point on that line has y = −2.

(4, -2)

Step 5. Put them together as an ordered pair.

Answer: They cross at (4, −2).

Example 5: Word problem

A parking garage charges $12 to park for any amount of time up to 8 hours. Write an equation for the cost C after h hours. What is the slope, and what does it mean?

Step 1. The cost is $12 no matter how many hours you park. So C is always 12.

C = 12

Step 2. Write the equation.

m = \frac{12 - 12}{8 - 0}

Step 3. Pick two points on the graph, such as (0, 12) and (8, 12). Put them into the slope formula.

m = \frac{0}{8}

Step 4. Subtract on the top and on the bottom.

m = 0

Step 5. Divide. Zero divided by any other number is 0.

Answer: $C = 12$ , for up to 8 hours. The graph is a horizontal line, so the slope is 0. The cost goes up by $0 for each extra hour.

Common Mistakes

Mistake	What it looks like	How to fix it
Using the wrong value	Horizontal line through (2, 5) written as $x = 2$	A horizontal line keeps y the same: $y = 5$ .
Mixing up the slopes	“The slope of $x = 3$ is 0.”	Vertical lines have undefined slope. Horizontal lines have slope 0.
Graphing in the wrong direction	Drawing $y = 4$ up and down through 4 on the x-axis	$y = 4$ crosses the y-axis at 4 and goes across.
Thinking it’s one point	“ $x = 3$ is the point (3, 0).”	$x = 3$ is a whole line of points like (3, 0), (3, 1) and (3, −8).

Horizontal and Vertical Lines in Slope-Intercept Form

A horizontal line fits the form $y = mx + b$ with a slope of 0. For example, $y = 0x + 4$ is the same as $y = 4$ .

A vertical line does not fit that form. It has no slope to put in for m, so you always write it as $x = a$ . You’ll use this form in the next lesson, Slope-Intercept Form.

A horizontal line and a vertical line always meet at a right angle, so they are perpendicular.

Practice Problems

Write each equation, or answer the question.

The horizontal line through (3, 8)
The vertical line through (−6, 1)
The line through (−4, 9) and (5, 9). What is its slope?
The line through (0, −3) and (0, 7). What is its slope?
What is the slope of $y = -7$ ?
Is $x = -1$ horizontal or vertical?
Where do $y = 6$ and $x = -2$ cross?
The line through (7, −4) that is parallel to the y-axis
The line through (−5, 2) that is parallel to the x-axis
A hot-air balloon floats at a steady height of 300 feet for 10 minutes. Write an equation for its height h after t minutes. What is the slope?

Show answers

A horizontal line keeps the same y-value

y = 8

The y-value of (3, 8) is 8.

A vertical line keeps the same x-value

x = -6

The x-value of (−6, 1) is −6.

y = 9

Both y-values are 9, so the line is horizontal.

m = \frac{9 - 9}{5 - (-4)} = \frac{0}{9} = 0

Find the slope.

x = 0

Both x-values are 0, so the line is vertical. It is the y-axis.

m = \frac{7 - (-3)}{0 - 0} = \frac{10}{0}

Find the slope.

You can’t divide by 0
The slope is undefined.

$y = -7$ has the same y-value at every point, so it is a horizontal line

m = 0

A horizontal line has slope 0.

x is always −1, and y can be anything
It is a vertical line.

Take the x-value from $x = -2$
Take the y-value from $y = 6$

(-2, 6)

Write the ordered pair.

They cross at (−2, 6).

A line parallel to the y-axis is vertical, so it keeps the same x-value

x = 7

The x-value of (7, −4) is 7.

A line parallel to the x-axis is horizontal, so it keeps the same y-value

y = 2

The y-value of (−5, 2) is 2.

10.

h = 300 \text{ for } 0 \le t \le 10

The height stays 300 feet the whole time.

m = \frac{300 - 300}{10 - 0} = \frac{0}{10} = 0

Use the points (0, 300) and (10, 300) to find the slope.

The slope is 0 feet per minute, because the height doesn’t change.

Want more practice? The Algebra 1 Unit 4 Practice Pack (Part 1) has 39 pages of problems on this and the other lessons in the unit, with step-by-step answers and a cheat sheet.

Horizontal and Vertical Lines FAQ

Is y = 5 a horizontal or vertical line?

The line $y = 5$ is horizontal. Every point on it has a y-value of 5, so it runs straight across.

What is the slope of a vertical line?

The slope of a vertical line is undefined. The run between any two points is 0, and you can’t divide by 0.

What is the equation of the x-axis?

The x-axis is the line $y = 0$ , and the y-axis is the line $x = 0$ .

Are horizontal and vertical lines perpendicular?

Yes. A horizontal line and a vertical line always cross at a right angle.

Summary

A horizontal line has the equation $y = b$ and a slope of 0.
A vertical line has the equation $x = a$ and an undefined slope.
The equation names the value that stays the same: y for horizontal lines, x for vertical lines.
The x-axis is $y = 0$ , and the y-axis is $x = 0$ .
A horizontal line and a vertical line cross at (a, b).

Previous lesson: Slope

Next lesson: Slope-Intercept Form

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