Parallel and Perpendicular Lines

Parallel Lines

Two lines are parallel if they have the same slope but different y-intercepts.

$m_1 = m_2 \quad \text{(parallel)}$

Parallel lines never intersect. In a system of equations, parallel lines mean no solution.

Example: $y = 3x + 5$ is parallel to $y = 3x - 2$ (both have slope 3).

Perpendicular Lines

Two lines are perpendicular if their slopes are negative reciprocals of each other.

$m_1 \times m_2 = -1 \quad \text{(perpendicular)}$

Equivalently: $m_2 = -\frac{1}{m_1}$

Perpendicular lines meet at a 90° angle.

Example: If a line has slope $\frac{2}{3}$, a perpendicular line has slope $-\frac{3}{2}$.

Example: If a line has slope $-4$, a perpendicular line has slope $\frac{1}{4}$.

Finding Slope from Any Equation Form

• Slope-intercept $y = mx + b$: slope is $m$
• Standard form $Ax + By = C$: slope is $-\frac{A}{B}$
• Point-slope $y - y_1 = m(x - x_1)$: slope is $m$

Writing Equations of Parallel Lines

To write the equation of a line parallel to a given line and passing through a specific point:

1. Find the slope of the given line.
2. Use the same slope (parallel = same slope).
3. Use point-slope form with the given point.

Example: Write the equation of the line parallel to $y = 2x + 5$ passing through $(3, 1)$.

Slope: $m = 2$

$y - 1 = 2(x - 3)$ $y = 2x - 5$

Writing Equations of Perpendicular Lines

1. Find the slope of the given line.
2. Take the negative reciprocal for the perpendicular slope.
3. Use point-slope form with the given point.

Example: Write the equation of the line perpendicular to $y = -\frac{1}{3}x + 4$ passing through $(2, 7)$.

Original slope: $-\frac{1}{3}$ → Perpendicular slope: $3$

$y - 7 = 3(x - 2)$ $y = 3x + 1$

Special Cases

• A horizontal line ($y = k$, slope $= 0$) is perpendicular to a vertical line ($x = k$, slope undefined).
• Two horizontal lines are parallel to each other.
• Two vertical lines are parallel to each other.

Tip 1: Negative Reciprocal Drill

Practise finding negative reciprocals instantly:

Original slope	Negative reciprocal
$2$	$-\frac{1}{2}$
$-3$	$\frac{1}{3}$
$\frac{4}{5}$	$-\frac{5}{4}$
$-\frac{1}{7}$	$7$
$1$	$-1$

Tip 2: Quick Check — Multiply the Slopes

To verify perpendicularity, multiply the two slopes. If the product is $-1$, they're perpendicular.

Tip 3: Convert to Slope-Intercept First

If the equation is in standard form, convert to $y = mx + b$ to easily identify the slope.

Tip 4: Don't Confuse Parallel with Perpendicular

Parallel = same slope. Perpendicular = negative reciprocal. The SAT sometimes tries to trick you by mixing these up in answer choices.

Tip 5: Look at the Graph

If a graph is provided, visually check: do the lines appear to meet at a right angle? Do they appear to never meet? Use this to guide your algebraic work.