$Slope and Linear Equations infographic - Rise over Run, Slope-Intercept, and Point-Slope Forms$

Click image to open full size

Math

Slope and Linear Equations

Rise over Run, Slope-Intercept, and Point-Slope Forms

Download PNG Save to Pinterest

Related Tools

Related Labs

Systems of Equations Lab

Related Worksheets

Related Cheat Sheets

Slope measures how steeply a line rises or falls. It is defined as the ratio of vertical change (rise) to horizontal change (run) between any two points on the line: m = (y₂ − y₁)/(x₂ − x₁). A positive slope means the line rises left to right; a negative slope means it falls; slope = 0 means horizontal; an undefined slope means vertical (division by zero). Because a line has constant slope everywhere, you only need two points to calculate it.

The slope-intercept form $y = mx + b$ is the most common way to write a linear equation. Here $m$ is the slope and $b$ is the y-intercept (the y-value where the line crosses the y-axis). This form makes graphing straightforward: start at $(0, b)$ and use the slope to find additional points. Point-slope form $y − y₁ = m(x − x₁)$ is useful when you know a point and the slope but not the y-intercept. Two lines with the same slope are parallel; perpendicular lines have slopes that are negative reciprocals of each other ( $m₁ \times m₂ = −1$ ).

Key Facts

Slope: $m = \frac{y₂ − y₁}{x₂ − x₁} = \frac{\text{rise}}{\text{run}}$
Slope-intercept form: $y = mx + b$ ( $m = \text{slope}$ , $b = \text{y-intercept}$ )
Point-slope form: $y − y₁ = m(x − x₁)$
Standard form: $Ax + By = C$ (useful for $x$ and $y$ intercepts)
Parallel lines: same slope ( $m₁ = m₂$ ); perpendicular lines: negative reciprocal slopes ( $m₁ \times m₂ = −1$ )
Horizontal line: slope = $0$ , equation $y = k$ ; vertical line: undefined slope, equation $x = k$

Vocabulary

Slope: The ratio of vertical change to horizontal change between two points on a line; measures steepness and direction.
Y-intercept: The point where a line crosses the y-axis (where $x = 0$ ); the value $b$ in $y = mx + b$ .
Slope-intercept form: The equation form $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept; directly shows the rate of change and starting value.
Linear equation: An equation whose graph is a straight line; all variables are to the first power and no variable is multiplied by another.
Rate of change: How much one quantity changes per unit change in another; for a linear function, this is constant and equal to the slope.

Common Mistakes to Avoid

Computing slope as $\frac{\text{run}}{\text{rise}}$ instead of $\frac{\text{rise}}{\text{run}}$ . Slope = $\frac{\Delta y}{\Delta x}$ (vertical over horizontal). Reversing this gives the reciprocal, not the slope.
Confusing the y-intercept with the x-intercept. The y-intercept ( $b$ in $y = mx + b$ ) is where the line crosses the y-axis (set $x = 0$ ). The x-intercept is where it crosses the x-axis (set $y = 0$ and solve for $x$ ).
Assuming any two points give a different slope for the same line. Slope is constant everywhere on a straight line - it doesn't matter which two points you choose.
Misapplying the perpendicular slope rule. Perpendicular slopes are negative reciprocals: if m = 2/3, the perpendicular slope is −3/2. Both the sign change and the flip are required.

Practice Questions

1 Find the slope and y-intercept of the line passing through (−2, 5) and (4, −1). Write the equation in slope-intercept form.
2 Line A has equation y = (3/4)x − 2. Write the equation of a line perpendicular to A passing through the point (0, 5).
3 A car travels at a constant speed. After 2 hours it has gone 120 km; after 5 hours it has gone 300 km. What is the slope of the distance-time graph and what does it represent physically?

Sign in to save