Problem 23 - Introductory and Intermediate Algebra for College Students 4th

Question

Use intercepts and a checkpoint to graph each equation. $$6 x-9 y=18$$

Step-by-Step Solution

Answer

The graph of the equation passing through the points X-intercept (3,0) and Y-intercept (0,-2). The point (1,1) is a checkpoint which does not satisfy the equation and therefore does not lie on the line.

Step 1: Find the x-intercept

Let y=0, we substitute y=0 into the equation and solve for x to get the x-intercept. So, we get \(6x - 9*0 = 18\), simplifies to \(6x = 18\), and upon dividing by 6 from both sides we get x=3. So, our x-intercept is (3,0).

Step 2: Find the y-intercept

Similarly, let x=0, then substitute x=0 into the equation and solve for y to find the y-intercept. Thus, \(6*0 - 9y = 18\), which simplifies to \(-9y = 18\), and upon dividing by -9 from both sides, we get y=-2. So, our y-intercept is (0,-2).

Step 3: Choose a checkpoint

Now, we choose a point not on the line made by the intercepts (i.e., neither (3,0) nor (0,-2)). The point (1,1) is a simple choice. Substituting x=1 and y=1 into the equation, we have \(6*1 - 9*1 = -3\), which is not equal to 18. Hence this point (1,1) does not lie on the line.

Step 4: Graph the equation

Using the x and y intercepts (3,0) and (0,-2), we plot these points on the graph. Then draw a line passing through the intercept points to represent the equation.

Key Concepts

X-Intercept

The x-intercept of a graph is where the line crosses the x-axis. To find it, you set the y-coordinate to 0 and solve the equation for x. In the equation 6x - 9y = 18, when we set y = 0, it simplifies to 6x = 18. Dividing both sides by 6 gives us x = 3, so the x-intercept is (3,0). This is a pivotal point since it establishes where the line begins on the horizontal axis.

When practicing with x-intercepts, always remember:

• Set y to zero before solving for x.
• The resulting point will always have the form (x,0).

Understanding x-intercepts is fundamental in graphing linear equations as they give us one of the two necessary points to draw the line of the equation on a graph.

Y-Intercept

In contrast to the x-intercept, the y-intercept is where a line crosses the y-axis. Here, you'll set the x-coordinate to 0 and solve for y. Taking our earlier example, 6x - 9y = 18, and setting x = 0 we get -9y = 18. Dividing both sides by -9, the solution is y = -2, which gives us the y-intercept as (0,-2).

The y-intercept is equally important as the x-intercept because it marks the start of the line on the vertical axis. Keep in mind to:

• Set x to zero before finding y.
• The point obtained will have a (0,y) form.

Understanding both intercepts is crucial as these are the common points used to sketch the graph of a linear equation.

Algebraic Graphing

Moving forward, algebraic graphing is a technique used to visually represent solutions of equations on a coordinate plane. Once we have our x and y-intercepts from the equation 6x-9y=18, we plot these points on a Cartesian plane.

For algebraic graphing, following these steps ensures accurate representation:

• Plot the x-intercept and y-intercept accurately on the graph.
• Use a straightedge to draw a line through these intercepts, which is your linear equation.
• If necessary, select an additional 'checkpoint' outside the intercepts to ensure the line is correct.

In our example, the checkpoint could verify if our line is correct, but since (1,1) doesn't satisfy the equation, it tells us this point isn't on our graphed line.

Linear Equations

Lastly, let's talk about linear equations. These equations form straight lines when graphed and typically look like ax + by = c, where a, b, and c are constants. They have a constant rate of change and no variable is raised to a power other than 1. By finding the x-intercept and y-intercept, you have all the necessary information to draw the line.

Key points to note about linear equations include:

• They produce straight lines on a graph.
• The slope-intercept form, y = mx + b, reveals the slope (m) and y-intercept (b) directly, which can also be useful in graphing the line quickly.

Our given equation of 6x - 9y = 18 is a classic linear equation and provides a straightforward example of how linear relationships are visually represented in graphing.