Positive Slope

Algebra-1

1. Fundamental Concepts

Definition: The rate of change is a measure of how much one quantity changes in relation to another quantity.
Positive Slope: A positive slope indicates that the value of the dependent variable increases as the independent variable increases.
Slope Formula: The slope \(m\) of a line is given by the formula \(m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\).

2. Key Concepts

Basic Rule: $$If \, m > 0, \, then \, the \, slope \, is \, positive.$$

Degree Preservation: The slope remains consistent for all points on a straight line.

Application: Used to analyze trends and predict outcomes in various fields such as economics, physics, and engineering.

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: \(m = \frac{{y_2 - y_1}}{{x_2 - x_1}} = \frac{{7 - 3}}{{4 - 2}} = \frac{4}{2} = 2\).

Validation: Substitute into the slope formula → Original: \(\frac{{7 - 3}}{{4 - 2}} = 2\); Simplified: \(2 = 2\) ✓

Example 2 (Intermediate)

Problem: Determine if the line with points \((-1, 5)\) and \((3, 9)\) has a positive slope.

Step-by-Step Solution:

Identify the coordinates: \((x_1, y_1) = (-1, 5)\) and \((x_2, y_2) = (3, 9)\).
Apply the slope formula: \(m = \frac{{y_2 - y_1}}{{x_2 - x_1}} = \frac{{9 - 5}}{{3 - (-1)}} = \frac{4}{4} = 1\).

Validation: Substitute into the slope formula → Original: \(\frac{{9 - 5}}{{3 - (-1)}} = 1\); Simplified: \(1 = 1\) ✓

4. Problem-Solving Techniques

Visual Strategy: Plot the points on a coordinate plane to visually determine the direction of the slope.
Error-Proofing: Double-check the subtraction in the numerator and denominator when calculating the slope.
Concept Reinforcement: Practice with different sets of points to understand the consistency of the slope formula.