Slope and Linear Relationships
Calculate and interpret slope as a rate of change, use slope-intercept and point-slope forms, and identify parallel and perpendicular lines.
Slope is arguably the most important concept in algebra. Every rate you encounter in real life — miles per hour, cost per item, interest rate per year, calories burned per minute — is a slope. Understanding slope lets you predict values, compare rates, and model any relationship where one quantity changes relative to another. It's the bridge from algebra to calculus and statistics.
- Calculate slope using rise over run or the slope formula
- Interpret slope as a rate of change in real-world contexts
- Write linear equations in slope-intercept form (y = mx + b) and point-slope form
- Identify parallel lines (equal slopes) and perpendicular lines (negative reciprocal slopes)
Rises left→right
Positive slope
Falls left→right
Negative slope
Horizontal line
Zero slope
Vertical line
No slope
Find the slope of the line through (1, 3) and (4, 9).
\[ m = \frac{9 - 3}{4 - 1} = \frac{6}{3} = 2 \]The slope is 2: for every 1 unit right, the line rises 2 units up.
Write the equation of a line with slope 3 passing through the point (2, 5).
Point-slope form:
\[ y - 5 = 3(x - 2) \]Convert to slope-intercept:
\[ y = 3x - 6 + 5 = 3x - 1 \]Equation: \(y = 3x - 1\). The y-intercept is −1; the line rises 3 for every 1 right.
A car uses 2.5 gallons of fuel per hour at highway speed. After 0 hours: 15 gallons remain. Model the fuel level \(F\) as a function of time \(t\), and find when the tank is empty.
- Slope = −2.5 (fuel decreasing), y-intercept = 15: \(F = -2.5t + 15\)
- Set F = 0: \(0 = -2.5t + 15\) → \(t = 6\)
The tank empties after 6 hours. The negative slope represents consumption rate.
- Find the slope through (−2, 1) and (4, 7).
- Write the equation of a line with slope −2 and y-intercept 5.
- Line A has slope 3/4. What slope is perpendicular to it?
▶ Show Answers
- \(m = \frac{7-1}{4-(-2)} = \frac{6}{6} =\) 1
- \(y = -2x + 5\)
- Negative reciprocal: −4/3
- Reversing rise and run: Slope = rise/run = (y₂−y₁)/(x₂−x₁), NOT (x₂−x₁)/(y₂−y₁).
- Inconsistent point ordering: If you put point 2 first in the numerator, put point 2 first in the denominator too. Be consistent.
- Perpendicular vs. parallel slopes: Parallel lines have equal slopes. Perpendicular lines have negative reciprocal slopes (flip and negate).
- Slope = rise/run = (y₂−y₁)/(x₂−x₁) — vertical change over horizontal change.
- Slope-intercept form y = mx + b: m tells steepness, b tells where it crosses the y-axis.
- Point-slope form y − y₁ = m(x − x₁): use when you have a point and a slope.
- Parallel lines share slope; perpendicular lines have slopes that are negative reciprocals.
Economists plot cost curves, revenue lines, and supply/demand graphs — all linear relationships interpreted through slope. A slope of $50/unit means each additional unit costs $50 to produce. A negative slope on a demand curve means higher prices lead to lower demand. Every trend line in a business presentation is a slope calculation dressed up in professional language.
Calculator Connection
Use the Slope Calculator to find slope between any two points. The Slope-Intercept Form Calculator converts equations to y = mx + b, and the Point-Slope Form Calculator writes the equation from a point and slope.
Interactive Diagram
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Slope and Linear Relationships - Quiz
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