Problem 98 - College Algebra Essentials

Question

Explain how to use intercepts to graph the general form of a line's equation.

Step-by-Step Solution

Answer

To use intercepts to graph the general form of a line's equation, first understand the line equation format, Axes + By = C, and intercept concepts. Then, calculate x and y intercepts by setting y and x to zero respectively and solving for the other variable. Plot these intercepts on the Cartesian plane, then draw a line through these points. This line represented the original equation.

Step 1: Define the Constitution of General Line Equation

The general form of a line's equation is \( Ax + By = C \), where A, B, and C are constants, and x, y are the variables. This equation represents a line on the Cartesian plane.

Step 2: Understand Intercept Concept

In the Cartesian plane, the x-intercept is the point where the line crosses the x-axis, and for that point, y = 0. The y-intercept is the point where the line crosses the y-axis, and for that point, x = 0.

Step 3: Find the Intercepts

For an equation in the general form, \( Ax + By = C \), to find the x-intercept, set y = 0 and solve for x. Thus, the x-intercept will be \( x = \frac{C}{A} \), if A ≠ 0. Similarly, for the y-intercept, set x = 0 and solve for y. Hence, the y-intercept will be \( y = \frac{C}{B} \), if B ≠ 0.

Step 4: Plot the Intercepts and Draw the Line

Plot these intercepts on the Cartesian plane, then draw a line through these points. This is the line represented by the given equation.