Building on Your Knowledge of Linear Equations and Inequalities
You’ve already learned how to create algebraic expressions, solve linear equations, and graph those equations on the coordinate plane. (If you need a refresher, revisit the earlier articles “Algebraic Expressions and Equations,” “Algebraic Inequalities: Comparing Values with Variables,” and “Graphing Linear Equations.”) In each of those lessons, the constant m in the equation y = mx + b appeared again and again. That constant is called the slope, and it represents one of the most important ideas in algebra: rate of change.
This article will show you what slope is, how to calculate it in different ways, and why it matters whenever two quantities are changing together.
1. What Is Slope?
On a graph, slope measures the steepness of a line. More precisely, it is the ratio of the vertical change (often called rise) to the horizontal change (often called run) between any two points on the line:
slope = m = rise/run = Δy/ΔxBecause a straight line has a constant steepness, this ratio is the same no matter which two points you pick.
2. Positive, Negative, Zero, and Undefined Slopes
Positive slope
If a line rises as you move to the right, m is positive. Example: the line from the article “Graphing Linear Equations” with equation y = 2x + 1 has m = 2.
Negative slope
If a line falls as you move right, m is negative. The line y = -x + 4 has m = –1.
Zero slope
A horizontal line such as y = 3 has m = 0. The rise is 0 because the y-value never changes.
Undefined slope
A vertical line such as x = -2 has no defined slope because the run is zero (division by zero).
3. Three Ways to Find Slope
A. From a Graph
Pick any two points on the line and count the rise and run.
Example: Suppose the line passes through (–2, –3) and (1, 3):
- Rise: from –3 to 3 is +6
- Run: from –2 to 1 is +3
So, m = 6/3 = 2
B. From Two Ordered Pairs
Use the formula:
m = (y₂ - y₁) / (x₂ - x₁)Example: Through (1, 3) and (5, 11):
m = (11 - 3) / (5 - 1) = 8 / 4 = 2C. From an Equation
Rewrite the equation in y = mx + b form. The coefficient of x is the slope.
Example: In 3y - 6x = 12, solve for y:
3y = 6x + 12 → y = 2x + 4So, m = 2, b = 4.
4. What Is Rate of Change?
In everyday language, rate of change tells how fast one quantity changes relative to another. In linear relationships, the numeric value of the slope is the rate of change.
Real-World Examples
| Situation | Independent Variable (x) | Dependent Variable (y) | Rate of Change (= Slope) |
|---|---|---|---|
| A taxi that charges $3 per mile | Miles traveled | Total cost | $3 per mile |
| A car moving at 60 mi/h | Hours | Miles driven | 60 miles per hour |
| Water in a tank draining 5 L/min | Minutes | Liters remaining | –5 liters per minute |
5. Interpreting Slope in Context
- Positive slope: Dependent variable increases as independent increases (e.g., earnings).
- Negative slope: Dependent variable decreases (e.g., decreasing fuel level).
- Zero slope: No change (e.g., a flat rate service).
Always include units: “The plant grows 2 cm per week.”
6. Connecting Slope to Inequalities
In the inequality article, you graphed x ≥ 4. With linear inequalities like y > 2x – 1, slope still controls the boundary line’s direction. The shading shows all points where the rate of change keeps y above the line.
7. Practice Problems
- Find the slope of the line through (–4, 1) and (2, –5).
- A phone company charges $25 per month plus $0.10 per text.
- Write an equation for total monthly cost C in terms of texts t.
- Identify and interpret the slope.
- A hiking trail descends 300 m over 2 km. What is the slope in m/km?
- Rewrite
4x – 2y = 6in slope–intercept form. Identify slope and y-intercept. - The temperature dropped from 20°C at 6 p.m. to 8°C at midnight. Find the average rate of change per hour.
8. Key Takeaways
- Slope = rise/run = rate of change
- You can find slope from a graph, two points, or an equation
- Units help describe real-world changes
- Positive, negative, zero, and undefined slopes describe line direction
- Slope helps you model and interpret real-world relationships
Mastering slope and rate of change prepares you for graphing linear inequalities, solving systems of equations, and analyzing real-world data. Keep practicing. Next, we will touch upon various forms of linear equations.
and soon you’ll recognize slopes and rates everywhere you look!