Express as Slope Intercept Form

Algebra-1

1. Fundamental Concepts

Definition: The slope-intercept form of a linear equation is expressed as $$y = mx + b$$ , 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

Basic Rule: $$y = mx + b$$

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

Application: Used to graph lines and solve real-world problems involving linear relationships

3. Examples

Example 1 (Basic)

Problem: Express the equation $$2y - 4x = 8$$ in slope-intercept form.

Step-by-Step Solution:

Rearrange the equation to isolate $$y$$ : $$2y = 4x + 8$$
Divide every term by 2: $$y = 2x + 4$$

Validation: Substitute $$x = 0$$ → Original: $$2(0) - 4(0) = 8$$ ; Simplified: $$y = 2(0) + 4 = 4$$ ✓

Example 2 (Intermediate)

Problem: Given two points $$(2, 5)$$ and $$(4, 9)$$ , find the equation of the line in slope-intercept form.

Step-by-Step Solution:

Calculate the slope $$m$$ : $$m = \frac{{9 - 5}}{{4 - 2}} = 2$$
Use one of the points to find $$b$$ : $$5 = 2(2) + b$$ , so $$b = 1$$
The equation is $$y = 2x + 1$$

Validation: Substitute $$x = 2$$ → Original: $$y = 5$$ ; Simplified: $$y = 2(2) + 1 = 5$$ ✓

4. Problem-Solving Techniques

Isolation Strategy: Always start by isolating $$y$$ on one side of the equation.
Point Verification: After finding the equation, verify it using one of the given points.
Graphical Interpretation: Use graphs to visualize the relationship between variables and understand the meaning of slope and intercept.