Special Cases: Vertical and Horizontal Lines
Vertical and horizontal lines are the special cases of linear equations covered in Grade 11 Algebra 1 enVision Chapter 2. Horizontal lines have slope m = 0 and take the form y = b — for example, y = 5 is a flat line at height 5. Vertical lines have undefined slope and take the form x = a — for example, x = 3 is a straight up-and-down line. All vertical lines are parallel to each other, all horizontal lines are parallel to each other, and every vertical line is perpendicular to every horizontal line. The usual m₁ · m₂ = -1 perpendicularity test does not apply here.
Key Concepts
Property Horizontal lines are flat, have a slope of $m = 0$, and take the equation format $y = b$. Vertical lines go straight up and down, have an undefined slope, and take the equation format $x = a$.
All vertical lines are parallel to each other, all horizontal lines are parallel to each other, and every vertical line is perfectly perpendicular to every horizontal line.
Examples The vertical line $x = 3$ is parallel to $x = 2$. The horizontal line $y = 5$ is perpendicular to the vertical line $x = 3$. Finding an equation: The vertical line passing through the point $( 2, 7)$ is simply $x = 2$. The horizontal line passing through that same point is $y = 7$.
Common Questions
Why does a vertical line have undefined slope?
Because slope = rise/run, and a vertical line has zero horizontal run. Dividing by zero is undefined.
What is the equation of a vertical line through (-2, 7)?
x = -2. Vertical lines have the form x = a, where a is the x-coordinate of any point on the line.
What is the equation of a horizontal line through (-2, 7)?
y = 7. Horizontal lines have the form y = b, where b is the y-coordinate of any point on the line.
Are all vertical lines parallel to each other?
Yes. All vertical lines (x = a) are parallel to each other because they all have undefined slope and never intersect.
Is x = 3 perpendicular to y = 5?
Yes. Every vertical line is perpendicular to every horizontal line, since their directions are at 90 degrees.
Why can you not use m₁ · m₂ = -1 to check if x = 3 and y = 5 are perpendicular?
Because that formula requires defined slopes for both lines. Since x = 3 has undefined slope, you use the geometric rule instead: all vertical and horizontal lines are perpendicular by definition.