Problem 83 - Introductory and Intermediate Algebra for College Students 4th

Question

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

Step-by-Step Solution

Answer

To graph a linear equation of the form \(A x + B y = 0\), first convert the equation to slope-intercept form. In this case, the equation becomes \(y = -\frac{A}{B}x\), the slope is -A/B, and the y-intercept is 0. Set up a graph with x and y axes, start at the origin and draw a line in the direction based on the slope. The line should be straight and pass straight through the origin.

Step 1: Convert to Slope-Intercept Form

Rearrange the given equation \(A x + B y = 0\) to slope-intercept form, \(y = mx + c\), by solving for \(y\). The reformulated equation becomes \(y = -\frac{A}{B}x\), as there is no constant term or \(c\).

Step 2: Identify Slope and Y-Intercept

In this case, the coefficient of \(x\) is the slope (m = -A/B) and there is no y-intercept because the line passes through the origin (c = 0).

Step 3: Set Up the Graph

Set up a graph with an x and y axis. Since there is no y-intercept, the line will pass through the origin (0,0).

Step 4: Plot the Line

Begin at the origin and use the slope to determine the direction of the line. If the slope is positive, the line will move up from left to right. If the slope is negative, the line will move down from left to right. Draw a straight line through these points.

Key Concepts

Slope-Intercept Form

The slope-intercept form is an essential concept in algebra for graphing and understanding linear equations. It is written as \(y = mx + b\), where \(m\) represents the slope of the line, and \(b\) specifies the y-intercept, which is the point where the line crosses the y-axis.

In practical terms, this form allows us to quickly identify the two key characteristics of a line on a graph: the slope and the starting point on the y-axis. Whenever you are given a linear equation in a different format, such as the standard form \(Ax + By = C\), a good initial step is to convert it into slope-intercept form. This is done by solving for \(y\) and putting the equation into the structure of \(y = mx + b\).

Let's apply this to the exercise where we have the equation \(Ax + By = 0\). Converting this to slope-intercept form requires isolating \(y\) on one side of the equation, leading to \(y = -\frac{A}{B}x\). Here, you'll notice there isn't a separate y-intercept term \(b\), which means the line goes through the origin, and \(b = 0\).

Slope of a Line

The slope is a numerical value that describes the steepness and the direction of a line on a graph. It's often denoted as \(m\) in the slope-intercept equation. A positive slope means that the line rises as it moves from left to right, whereas a negative slope indicates that the line falls.

The formula for slope is \(m = \frac{{rise}}{{run}}\), which is the vertical change divided by the horizontal change between any two points on the line. In the given exercise, we find the slope by rearranging the equation to get \(y = -\frac{A}{B}x\). Here, the slope \(m = -\frac{A}{B}\), telling us the line will decline as we move to the right if \(A\) and \(B\) are of opposite signs.

When graphing the line, you can use the slope to move from one point to the next. Starting from the y-intercept or another known point, move up or down by the rise value and then right or left by the run value, depending on the sign of the slope. This will give you another point through which the line will pass, helping you draw it accurately.

Y-Intercept

The y-intercept is where a line crosses the y-axis of a graph, and it's critically important for graphing and understanding the starting point of a line. In the slope-intercept form \(y = mx + b\), the y-intercept is represented by the \(b\) value. A line that has a y-intercept of, say, 3 would cross the y-axis at the point \((0,3)\).

In our exercise example, we're given a special case where the line crosses the origin. Since the equation does not have an independent constant term, the y-intercept is \(0\), resulting in the point \((0,0)\). This means that when you begin to graph, you will start plotting at the origin and use the slope to find other points on the line.

Knowing the y-intercept allows you to immediately place a point on the graph and is especially helpful as a starting reference point. Even if a line has no y-intercept term, as in this exercise, recognizing that the y-intercept is simply the origin simplifies the graphing process.