Problem 82 - Introductory Algebra for College Students

Question

Explain how to graph \(A x+B y=C\) if \(C\) is not equal to zero.

Step-by-Step Solution

Answer

First set \(y = 0\) to find the x-intercept (\(C/A\),0), then set \(x = 0\) to find the y-intercept (0, \(C/B\)). Finally, plot these points on a graph, and connect them with a straight line.

Step 1: Understanding the Equation

Given the equation \(A x + B y = C\), this standard form of a linear equation simplifies the process of finding the x and y intercepts. X-intercepts occur when \(y = 0\), and Y-intercepts occur when \(x = 0\). These intercepts are crucial as they provide two points through which the line passes. Before moving ahead, also note that \(C\) ≠ \(0\) as per the problem statement.

Step 2: Finding the X-Intercept

To find the x-intercept, set \(y = 0\) in our given equation and then solve for \(x\). This gives: \(Ax + B(0) = C\). Simplifying this gives \(x = C/A\).

Step 3: Finding the Y-Intercept

To find the y-intercept, set \(x = 0\) in our given equation and then solve for \(y\). This gives: \(A(0) + By = C\). Simplifying this gives \(y = C/B\).

Step 4: Plotting the Line

Now that we have the x-intercept (\(C/A\),0) and the y-intercept (0, \(C/B\)), plot these two points on the coordinate plane. Draw a line through these two points; this is the graph of the given equation \(A x + B y = C\).