Graph Standard Form - Algebra-1

1. Fundamental Concepts

Definition: The standard form of a linear equation is given by $$Ax + By = C$$ , where $$A$$ , $$B$$ , and $$C$$ are constants, and $$x$$ and $$y$$ are variables.
Graphing: To graph a linear equation in standard form, find the x-intercept by setting $$y = 0$$ and solving for $$x$$ , and find the y-intercept by setting $$x = 0$$ and solving for $$y$$ .
Slope: The slope of the line can be found using the formula $$m = -\frac{A}{B}$$ .

2. Key Concepts

Intercepts Method: To find the x-intercept, set $$y = 0$$ and solve for $$x$$ . To find the y-intercept, set $$x = 0$$ and solve for $$y$$ .

Slope Calculation: The slope $$m$$ of the line is given by $$m = -\frac{A}{B}$$ .

Graphing Technique: Plot the intercepts and draw a line through them to graph the equation.

3. Examples

Example 1 (Basic)

Problem: Graph the equation $$2x + 3y = 6$$ .

Step-by-Step Solution:

Find the x-intercept by setting $$y = 0$$ : $$2x + 3(0) = 6 \Rightarrow x = 3$$ .
Find the y-intercept by setting $$x = 0$$ : $$2(0) + 3y = 6 \Rightarrow y = 2$$ .
Plot the points $$(0, 2)$$ and $$(3, 0)$$ and draw a line through them.

Validation: Substituting $$x = 3$$ and $$y = 0$$ into the original equation confirms the x-intercept. Similarly, substituting $$x = 0$$ and $$y = 2$$ confirms the y-intercept.

Example 2 (Intermediate)

Problem: Graph the equation $$-4x + 5y = 20$$ .

Step-by-Step Solution:

Find the x-intercept by setting $$y = 0$$ : $$-4x + 5(0) = 20 \Rightarrow x = -5$$ .
Find the y-intercept by setting $$x = 0$$ : $$-4(0) + 5y = 20 \Rightarrow y = 4$$ .
Plot the points $$(-5, 0)$$ and $$(0, 4)$$ and draw a line through them.

Validation: Substituting $$x = -5$$ and $$y = 0$$ into the original equation confirms the x-intercept. Similarly, substituting $$x = 0$$ and $$y = 4$$ confirms the y-intercept.

4. Problem-Solving Techniques

Intercept Method: Always start by finding the x-intercept and y-intercept to plot the line accurately.
Slope Calculation: Use the slope formula $$m = -\frac{A}{B}$$ to understand the direction and steepness of the line.
Verification: Substitute the intercepts back into the original equation to ensure accuracy.