The concept of slope, rate of change

Algebra-1

1. Fundamental Concepts

Definition: The slope of a line is a measure of its steepness and direction, calculated as the ratio of the vertical change (rise) to the horizontal change (run).
Rate of Change: The slope represents the rate of change between two variables in a linear relationship.
Slope Formula: The slope \(m\) of a line passing through points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\).

2. Key Concepts

Slope-Intercept Form: $$y = mx + b$$ where \(m\) is the slope and \(b\) is the y-intercept.

Positive Slope: Indicates an upward trend from left to right.

Negative Slope: Indicates a downward trend from left to right.

3. Examples

Example 1 (Basic)

Problem: Find the slope of the line passing through the points \((2, 3)\) and \((5, 9)\).

Step-by-Step Solution:

Identify the coordinates: \((x_1, y_1) = (2, 3)\), \((x_2, y_2) = (5, 9)\).
Apply the slope formula: \(m = \frac{{y_2 - y_1}}{{x_2 - x_1}} = \frac{{9 - 3}}{{5 - 2}} = \frac{6}{3} = 2\).

Validation: Substitute into the slope formula with different points on the same line to verify consistency.

Example 2 (Intermediate)

Problem: Write the equation of a line with a slope of \(4\) that passes through the point \((1, 2)\).

Step-by-Step Solution:

Use the slope-intercept form \(y = mx + b\). Here, \(m = 4\).
Substitute the point \((1, 2)\) into the equation to find \(b\): \(2 = 4(1) + b\), so \(b = -2\).
The equation of the line is \(y = 4x - 2\).

Validation: Check if the point \((1, 2)\) satisfies the equation \(y = 4x - 2\).

4. Problem-Solving Techniques

Graphical Method: Plot the points and draw the line to visually determine the slope.
Algebraic Method: Use the slope formula and substitute known values to solve for unknowns.
Verification: Always check your solution by substituting back into the original problem or using another point on the line.