Problem 83 - Introductory Algebra for College Students

Question

Explain how to graph a linear equation of the form \(A x+B y=0\).

Step-by-Step Solution

Answer

Put the linear equation \(A x+B y=0\) in slope-intercept form \(y = mx + c\), identify the slope (\(m=-\frac{A}{B}\)) and the y-intercept (\(c = 0\)), then graph the equation by plotting the y-intercept as the first point and moving according to the slope to plot additional points, finally drawing a line through the points.

Step 1: Understanding the General Form

A linear equation may be written in general form as \(A x + B y = 0\), where A and B are real numbers.

Step 2: Rearranging the Equation

Rearrange the equation into slope-intercept form \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. In this case, the equation will be \(y = -\frac{A}{B}x + 0\), which means \(m = -\frac{A}{B}\) and \(c = 0\).

Step 3: Identifying Slope and y-intercept

Identify the slope (\(m\)) and the y-intercept (\(c\)). The slope is \(-\frac{A}{B}\) and the y-intercept is 0. These two values will allow to graph the linear equation.

Step 4: Graphing the Line

Plot the y-intercept (0) as the first point on the y-axis. Then, use the slope \(-\frac{A}{B}\) to get another point by moving right if \(-\frac{A}{B}\) is positive or left if it's negative by 1 unit (for the x-component), and moving up if \(-\frac{A}{B}\) is positive or down if it's negative by |\(-\frac{A}{B}\)| units (for the y-component). One may repeat the process to get more points. Then, draw a straight line through the points plotted on the graph.