Find the Slope and Y-Intercept

Algebra-1

1. Fundamental Concepts

Definition: The slope-intercept form of a linear equation is given by $$y = mx + b$$ , where $$m$$ is the slope and $$b$$ is the y-intercept.
Slope (m): Represents the steepness of the line; calculated as the change in y divided by the change in x ( $$m = \frac{{\Delta y}}{{\Delta x}}$$ ).
Y-Intercept (b): The point where the line crosses the y-axis; when $$x = 0$$ , $$y = b$$ .

2. Key Concepts

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

Graphical Interpretation: The slope $$m$$ indicates the rise over run, and the y-intercept $$b$$ is the starting point on the y-axis.

Application: Used to model real-world scenarios such as cost functions, distance-time relationships, etc.

3. Examples

Example 1 (Basic)

Problem: Find the slope and y-intercept of the line represented by the equation $$y = 3x + 4$$ .

Step-by-Step Solution:

The equation is already in slope-intercept form $$y = mx + b$$ .
Identify the slope $$m$$ and y-intercept $$b$$ : $$m = 3$$ and $$b = 4$$ .

Validation: The slope is 3 and the y-intercept is 4, which matches the given equation.

Example 2 (Intermediate)

Problem: Given two points $$(2, 5)$$ and $$(4, 9)$$ , find the slope and y-intercept of the line passing through these points.

Step-by-Step Solution:

Calculate the slope using the formula $$m = \frac{{y_2 - y_1}}{{x_2 - x_1}}$$ : $$m = \frac{{9 - 5}}{{4 - 2}} = \frac{4}{2} = 2$$ .
Use one of the points to find the y-intercept. Using $$(2, 5)$$ : $$5 = 2(2) + b$$ , so $$b = 1$$ .

Validation: Substitute $$x = 2$$ into the equation $$y = 2x + 1$$ to get $$y = 5$$ , which matches the given point.

4. Problem-Solving Techniques

Visual Strategy: Plot the points and draw the line to visually identify the slope and y-intercept.
Error-Proofing: Double-check calculations by substituting values back into the equation.
Concept Reinforcement: Practice with different types of problems to solidify understanding.