Express as Point Slope Form

Algebra-1

1. Fundamental Concepts

Definition: The point-slope form of a linear equation is given by $$y - y_1 = m(x - x_1)$$, where $$m$$ is the slope and $$(x_1, y_1)$$ is a point on the line.
Slope: The slope $$m$$ represents the rate of change of $$y$$ with respect to $$x$$.
Point: Any point $$(x_1, y_1)$$ that lies on the line can be used in the formula.

2. Key Concepts

Basic Rule: $$y - y_1 = m(x - x_1)$$

Slope Calculation: The slope $$m$$ can be calculated using two points $$(x_1, y_1)$$ and $$(x_2, y_2)$$ as $$m = \frac{y_2 - y_1}{x_2 - x_1}$$

Application: Used to write the equation of a line when the slope and a point are known

3. Examples

Example 1 (Basic)

Problem: Write the equation of the line with slope $$m = 3$$ passing through the point $$(2, 5)$$.

Step-by-Step Solution:

Substitute $$m = 3$$, $$x_1 = 2$$, and $$y_1 = 5$$ into the point-slope form: $$y - 5 = 3(x - 2)$$
Simplify the equation: $$y - 5 = 3x - 6$$
Rearrange to get the standard form: $$y = 3x - 1$$

Validation: Substitute $$x = 2$$ → Original: $$y = 5$$; Simplified: $$y = 3(2) - 1 = 5$$ ✓

Example 2 (Intermediate)

Problem: Write the equation of the line passing through the points $$(1, 3)$$ and $$(4, 9)$$.

Step-by-Step Solution:

Calculate the slope $$m = \frac{9 - 3}{4 - 1} = \frac{6}{3} = 2$$
Use either point for $$(x_1, y_1)$$. Using $$(1, 3)$$: $$y - 3 = 2(x - 1)$$
Simplify the equation: $$y - 3 = 2x - 2$$
Rearrange to get the standard form: $$y = 2x + 1$$

Validation: Substitute $$x = 1$$ → Original: $$y = 3$$; Simplified: $$y = 2(1) + 1 = 3$$ ✓

4. Problem-Solving Techniques

Visual Strategy: Plot the points and draw the line to visualize the slope and intercepts.
Error-Proofing: Double-check calculations and ensure the slope is correctly applied in the formula.
Concept Reinforcement: Practice converting between different forms of linear equations to reinforce understanding.