Negative 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. When the rate of change is negative, it indicates that one quantity decreases as the other increases.
Negative Slope: A negative slope represents a downward trend on a graph, indicating a decrease in the dependent variable as the independent variable increases.
Slope Formula: The slope \(m\) of a line is given by the formula \(m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\), where \((x_1, y_1)\) and \((x_2, y_2)\) are points on the line.

Interpretation: A negative slope (\(m < 0\)) indicates an inverse relationship between variables.

Graphical Representation: On a coordinate plane, a negative slope appears as a line sloping downwards from left to right.

Real-World Application: Examples include depreciation of assets over time or decreasing temperature with increasing altitude.

Problem: Find the slope of the line passing through the points \((2, 5)\) and \((4, 1)\).

Use the slope formula: \(m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\)
Substitute the given points: \(m = \frac{{1 - 5}}{{4 - 2}} = \frac{{-4}}{{2}} = -2\)

Validation: The slope \(m = -2\) indicates a negative rate of change.

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

Rearrange the equation into slope-intercept form \(y = mx + b\): \(3y = -2x + 6\)
Solve for \(y\): \(y = -\frac{2}{3}x + 2\)
The slope \(m\) is \(-\frac{2}{3}\).

Validation: The slope \(m = -\frac{2}{3}\) confirms a negative rate of change.

Graphical Method: Plot the points and draw a line to visually determine the slope direction.
Algebraic Method: Use the slope formula to calculate the exact value of the slope.
Contextual Understanding: Relate the slope to real-world scenarios to understand its significance.