Lesson 5: Properties of Lines

New Concept

This lesson reveals how a line's slope is the key to its identity. We'll use slope to define parallel ($m_1=m_2$) and perpendicular ($m_1m_2=-1$) relationships, and to write equations for horizontal and vertical lines.

What’s next

Now you have the foundational rules. Next, you’ll work through interactive examples and practice problems to apply these properties and master the slope formula.

Property

Two lines are parallel if their slopes are equal, that is, if

or if both lines are vertical.

Examples

• The lines given by $y = 3x - 2$ and $y = 3x + 5$ are parallel because both have a slope of $m=3$.
• To check if $2x + y = 5$ and $4x + 2y = 12$ are parallel, we find their slopes. The first equation is $y = -2x + 5$ ($m_1 = -2$). The second is $y = -2x + 6$ ($m_2 = -2$). Since $m_1 = m_2$, the lines are parallel.
• The vertical lines $x = 4$ and $x = -2$ are parallel. They both have undefined slopes but will never intersect.

Explanation

Parallel lines are like train tracks—they run in the exact same direction and never meet. This means they must have the identical steepness, or slope. If their slopes were different, they would eventually cross.

Property

Two lines are perpendicular if the product of their slopes is $-1$, that is, if

or if one of the lines is horizontal and one is vertical. The slope of one perpendicular line is the negative reciprocal of the other:

Examples

• A line with slope $m_1 = 5$ is perpendicular to a line with slope $m_2 = -\frac{1}{5}$ because their product is $5 \times (-\frac{1}{5}) = -1$.
• The lines $y = \frac{3}{4}x + 2$ and $y = -\frac{4}{3}x - 1$ are perpendicular. Their slopes are $m_1 = \frac{3}{4}$ and $m_2 = -\frac{4}{3}$, and their product is $(\frac{3}{4})(-\frac{4}{3}) = -1$.
• The line $y = 5$ is horizontal (slope is 0), and the line $x = 1$ is vertical (slope is undefined). These lines are perpendicular.

Explanation

Perpendicular lines intersect at a perfect right angle, like the corner of a book. Their slopes are opposites in two ways: one is positive while the other is negative, and their fractions are flipped (reciprocals).

Property

1. The equation of the horizontal line passing through $(0, b)$ is

2. The equation of the vertical line passing through $(a, 0)$ is

Examples

• The equation of the horizontal line passing through the point $(4, 7)$ is $y = 7$. Every point on this line has a y-coordinate of 7.
• The equation of the vertical line that contains the point $(-2, 5)$ is $x = -2$. Every point on this line has an x-coordinate of -2.
• The x-axis is a horizontal line with the equation $y=0$, and the y-axis is a vertical line with the equation $x=0$.

Explanation

For a horizontal line, every point has the same height (y-value), so its equation is just $y$ equals that constant value. For a vertical line, every point is the same distance left or right (x-value), so its equation is $x$ equals that constant value.

Property

The slope of the line joining points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$ is

Examples

• To find the slope of the line through $(1, 2)$ and $(4, 8)$, we let $(x_1, y_1) = (1, 2)$ and $(x_2, y_2) = (4, 8)$. The slope is $m = \frac{8 - 2}{4 - 1} = \frac{6}{3} = 2$.
• The slope of the line containing points $(-2, 5)$ and $(3, -5)$ is calculated as $m = \frac{-5 - 5}{3 - (-2)} = \frac{-10}{5} = -2$.
• For the points $(5, -3)$ and $(-1, -1)$, the slope is $m = \frac{-1 - (-3)}{-1 - 5} = \frac{2}{-6} = -\frac{1}{3}$. It doesn't matter which point you choose as first or second.

Explanation

This formula is a precise way to calculate 'rise over run.' It finds the vertical change (the 'rise,' $y_2 - y_1$) and divides it by the horizontal change (the 'run,' $x_2 - x_1$) between any two points on a line.