Standard Form - Find x and y Intercepts

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.
x-Intercept: The x-intercept is the point where the line crosses the x-axis (where $$y = 0$$ ).
y-Intercept: The y-intercept is the point where the line crosses the y-axis (where $$x = 0$$ ).

2. Key Concepts

Finding x-Intercept: Set $$y = 0$$ in the equation $$Ax + By = C$$ and solve for $$x$$ .

Finding y-Intercept: Set $$x = 0$$ in the equation $$Ax + By = C$$ and solve for $$y$$ .

Graphical Interpretation: The intercepts help in plotting the line on a coordinate plane.

3. Examples

Example 1 (Basic)

Problem: Find the x and y intercepts of the equation $$2x + 3y = 6$$ .

Step-by-Step Solution:

To find the x-intercept, set $$y = 0$$ : $$2x + 3(0) = 6 \Rightarrow 2x = 6 \Rightarrow x = 3$$
To find the y-intercept, set $$x = 0$$ : $$2(0) + 3y = 6 \Rightarrow 3y = 6 \Rightarrow y = 2$$

Validation: Substituting $$x = 3$$ and $$y = 2$$ into the original equation confirms the intercepts.

Example 2 (Intermediate)

Problem: Find the x and y intercepts of the equation $$4x - 5y = 20$$ .

Step-by-Step Solution:

To find the x-intercept, set $$y = 0$$ : $$4x - 5(0) = 20 \Rightarrow 4x = 20 \Rightarrow x = 5$$
To find the y-intercept, set $$x = 0$$ : $$4(0) - 5y = 20 \Rightarrow -5y = 20 \Rightarrow y = -4$$

Validation: Substituting $$x = 5$$ and $$y = -4$$ into the original equation confirms the intercepts.

4. Problem-Solving Techniques

Substitution Method: Always substitute zero for one variable to find the intercept with respect to the other variable.
Check Consistency: After finding the intercepts, substitute them back into the original equation to ensure they satisfy it.
Graphical Representation: Plotting the intercepts can help visualize the line and confirm the solution graphically.