1. What is Slope?

Slope measures the steepness of a line between two points. It represents the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line.

2. How to Calculate Slope

The slope (m) is calculated using the formula:

Where:

• \( (x_1, y_1) \) — Coordinates of the first point
• \( (x_2, y_2) \) — Coordinates of the second point
• \( m \) — Slope of the line (unitless or % if converted)

Explanation: The numerator represents the vertical change (rise), while the denominator represents the horizontal change (run).

3. Interpretation of Slope Values

Positive Slope: Line rises from left to right
Negative Slope: Line falls from left to right
Zero Slope: Horizontal line
Undefined Slope: Vertical line (division by zero)
Steepness: The greater the absolute value, the steeper the line

4. Practical Applications

Examples: Calculating roof pitch, road grades, wheelchair ramps, economic trends, physics problems, and more.

5. Frequently Asked Questions (FAQ)

Q1: What does a slope of 1 mean?
A: A slope of 1 means for every 1 unit increase in x, y increases by 1 unit (45° angle if axes are scaled equally).

Q2: How do you convert slope to degrees?
A: Angle = arctan(slope). For example, a slope of 1 equals 45°.

Q3: What's the difference between slope and gradient?
A: In mathematics, they're often synonymous. In some fields, gradient may refer to the slope in percentage terms.

Q4: Can slope be negative?
A: Yes, negative slope indicates the line is decreasing from left to right.

Q5: What does an undefined slope mean?
A: An undefined slope occurs when x₁ = x₂ (vertical line), resulting in division by zero.