Slopes, Equations of Lines, and Relationships Between Lines in the Cartesian Plane

Slopes and Equations of Lines

Determining the Slope of a Line

The slope of a line measures its steepness and direction. It is calculated as the ratio of the change in y to the change in x between two points on the line.

• Formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Example: For points $(-2,4)$ and $(3,-6)$: $m = \frac{-6 - 4}{3 - (-2)} = \frac{-10}{5} = -2$

Equation of a Line Using Point-Slope Form

The point-slope form is useful for writing the equation of a line when you know its slope and a point on the line.

• Formula: $y - y_1 = m(x - x_1)$

• Example: Using $m = -2$ and point $(-2,4)$: $y - 4 = -2(x + 2)$

Equation of a Line in Standard Form

The standard form of a linear equation is $Ax + By = C$, where $A$, $B$, and $C$ are integers and $A > 0$.

• Example: Convert $y - 4 = -2(x + 2)$ to standard form: $y - 4 = -2x - 4$ $y = -2x$ $2x + y = 0$

Finding Slope and y-intercept from Standard Form

To find the slope and y-intercept from an equation in standard form, solve for $y$ to get the slope-intercept form $y = mx + b$.

• Example: For $-3x + 5y = 20$: $5y = 3x + 20$ $y = \frac{3}{5}x + 4$ Slope: $\frac{3}{5}$ y-intercept: $4$

Parallel and Perpendicular Lines

Parallel Lines

Two distinct nonvertical lines in the Cartesian plane are parallel if and only if they have the same slope.

• Theorem: Two distinct nonvertical lines are parallel if and only if their slopes are equal.

• Example: $y = \frac{3}{2}x - 1$ and $4x + 6y = 12$ Convert $4x + 6y = 12$ to slope-intercept form: $6y = -4x + 12$ $y = -\frac{2}{3}x + 2$ Slope of first line: $\frac{3}{2}$ Slope of second line: $\frac{3}{2}$ Conclusion: The lines are parallel because their slopes are equal.

Perpendicular Lines

Two nonvertical lines in the Cartesian plane are perpendicular if and only if the product of their slopes is $-1$ (i.e., their slopes are negative reciprocals).

• Theorem: Two nonvertical lines are perpendicular if and only if the product of their slopes is $-1$.

• Formula: If $m_1$ and $m_2$ are slopes, then $m_1 \cdot m_2 = -1$

• Example: $3x - 6y = -12$ and $2x + y = 4$ Find slopes: $3x - 6y = -12$ $-6y = -3x - 12$ $y = \frac{1}{2}x + 2$ Slope $m_1 = \frac{1}{2}$ $2x + y = 4$ $y = -2x + 4$ Slope $m_2 = -2$ $m_1 \cdot m_2 = \frac{1}{2} \cdot (-2) = -1$ Conclusion: The lines are perpendicular.

Summary Table: Parallel and Perpendicular Lines

Relationship	Condition on Slopes
Parallel	$m_1 = m_2$
Perpendicular	$m_1 \cdot m_2 = -1$
Neither	Slopes are not equal and not negative reciprocals

Determining Whether Two Lines Are Parallel, Perpendicular, or Neither

To determine the relationship between two lines, compare their slopes:

• Parallel: Slopes are equal.

• Perpendicular: Slopes are negative reciprocals.

• Neither: Slopes are neither equal nor negative reciprocals.

Examples

• Example 1: $3x - y = 4$ and $x + 3y = 7$ $3x - y = 4$ $-y = -3x + 4$ $y = 3x - 4$ Slope $m_1 = 3$ $x + 3y = 7$ $3y = -x + 7$ $y = -\frac{1}{3}x + \frac{7}{3}$ Slope $m_2 = -\frac{1}{3}$ $m_1 \cdot m_2 = 3 \cdot (-\frac{1}{3}) = -1$ Conclusion: Perpendicular

• Example 2: $y = \frac{1}{2}x + 3$ and $x + 2y = 1$ $x + 2y = 1$ $2y = -x + 1$ $y = -\frac{1}{2}x + \frac{1}{2}$ Slope $m_2 = -\frac{1}{2}$ $m_1 = \frac{1}{2}$ $m_1 \neq m_2$ and $m_1 \cdot m_2 = -\frac{1}{4} \neq -1$ Conclusion: Neither

• Example 3: $x = -1$ and $x = 2$ Both are vertical lines (undefined slope). Conclusion: Parallel

Additional Tips

• Vertical Lines: Any two distinct vertical lines are parallel to each other.

• Horizontal and Vertical Lines: Any horizontal line is perpendicular to any vertical line.

Summary of Key Concepts

• Slope: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Point-Slope Form: $y - y_1 = m(x - x_1)$

• Standard Form: $Ax + By = C$

• Parallel Lines: $m_1 = m_2$

• Perpendicular Lines: $m_1 \cdot m_2 = -1$

Additional info: These notes cover foundational concepts in College Algebra related to linear equations, slopes, and geometric relationships between lines in the Cartesian plane.