Find the slope of a graph

Algebra-1

1. Fundamental Concepts

Definition: The slope of a graph is a measure of its steepness, indicating the rate at which one variable changes with respect to another.
Formula: The slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\).
Interpretation: A positive slope indicates an upward trend, while a negative slope indicates a downward trend.

2. Key Concepts

Slope Calculation: $$m = \frac{{y_2 - y_1}}{{x_2 - x_1}}$$

Graphical Interpretation: The slope is the ratio of the vertical change (rise) to the horizontal change (run).

Application: Used in various fields such as physics, economics, and engineering to analyze trends and rates of change.

3. Examples

Example 1 (Basic)

Problem: Find the slope of the line passing through the points \((2, 3)\) and \((4, 7)\).

Step-by-Step Solution:

Identify the coordinates: \((x_1, y_1) = (2, 3)\) and \((x_2, y_2) = (4, 7)\).
Apply the slope formula: \(m = \frac{{7 - 3}}{{4 - 2}}\).
Calculate the slope: \(m = \frac{{4}}{{2}} = 2\).

Validation: Substitute the points into the slope formula to ensure consistency.

Example 2 (Intermediate)

Problem: Determine the slope of the line represented by the equation \(2x - 3y + 6 = 0\).

Step-by-Step Solution:

Rearrange the equation into slope-intercept form \(y = mx + b\): \(2x - 3y + 6 = 0 \Rightarrow -3y = -2x - 6 \Rightarrow y = \frac{2}{3}x + 2\).
The coefficient of \(x\) gives the slope: \(m = \frac{2}{3}\).

Validation: Check that the rearranged equation matches the original.

4. Problem-Solving Techniques

Visual Strategy: Plot the points on a coordinate plane to visualize the slope.
Error-Proofing: Double-check calculations by substituting values back into the slope formula.
Concept Reinforcement: Practice with different types of problems to reinforce understanding.