Problem 30 - Introductory and Intermediate Algebra for College Students 4th

Question

Use intercepts and a checkpoint to graph each equation. $$2 x=3 y+6$$

Step-by-Step Solution

Answer

The graph of the equation 2x = 3y + 6 cuts the x-axis at (3,0) and the y-axis at (0,-2). Checkpoint is at (1, -4/3).

Step 1: Rewrite the equation in slope-intercept form.

Rearranging the given equation, we get \(y = \frac{2}{3}x - 2\)

Step 2: Identify the y-intercept.

From the rearranged equation, the y-intercept (b) is -2. So, one point on the graph will be (0, -2).

Step 3: Identify the x-intercept.

The x-intercept occurs when y = 0. Substituting y = 0 into the equation, we get \(0 = \frac{2}{3}x - 2 \Rightarrow x = 3\). So another point on the graph is (3, 0)

Step 4: Find a suitable checkpoint.

A convenient checkpoint could be when x = 1. Replacing x = 1 in the rearranged equation gives \(y = \frac{2}{3} - 2 = -\frac{4}{3}\). So a checkpoint on the graph is (1, -4/3).

Step 5: Plot the graph.

Plot the y-intercept (0, -2), the x-intercept (3, 0), and the checkpoint (1, -4/3). Draw a straight line through these points to create the graph of the equation.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form of a line is crucial when you are graphing linear equations. In mathematics, the slope-intercept form is expressed as \( y = mx + b \), where \( m \) stands for the slope of the line, and \( b \) represents the y-intercept, which is the point where the line crosses the y-axis. To make it easier to graph a line, you can rearrange almost any linear equation into this form.

For instance, if you start with an equation like \( 2x = 3y + 6 \), you can manipulate it into the slope-intercept form by isolating y on one side. This results in \( y = \frac{2}{3}x - 2 \). Now, with the equation in this format, you can immediately identify the slope feature of the graph as \( \frac{2}{3} \), which defines how steep the line is and in which direction it goes. Overall, mastering this form simplifies graphing significantly and helps you visualize linear relationships easily.

Y-Intercept

The y-intercept of a line is simply the point where the line crosses the y-axis. It occurs when the value of x is zero. In the slope-intercept form, \( y = mx + b \), the y-intercept is represented by \( b \). Knowing the y-intercept gives you the starting point to graph a line.

For instance, in the earlier example where the equation was rewritten as \( y = \frac{2}{3}x - 2 \), the y-intercept is -2. This tells you that when x is zero, y will be -2, so you can mark this point as (0, -2) on the graph. This is one of your key points from which you'll draw the entire line, and it serves as a foundation for the linear equation's graphical representation.

X-Intercept

Conversely, the x-intercept is where the graph crosses the x-axis, and it happens when y is zero.

Finding the X-Intercept

To find it from the slope-intercept form, set \( y \) to zero and solve for \( x \). From the example equation \( y = \frac{2}{3}x - 2 \), setting \( y \) to 0 gives you \( 0 = \frac{2}{3}x - 2 \), which results in \( x = 3 \) after solving. So the x-intercept is at (3, 0) on the graph.

You can think of the x-intercept as where the popularity or usage of your favorite app plateaus, hitting a point where it no longer increases or decreases; it's a specific kind of 'milestone' on the graph representing a moment of equilibrium.

Checkpoint in Graphing

Using a 'checkpoint' is a reliable method to verify the accuracy of your graph.

Choosing a Checkpoint

It’s simply an additional point that lies on the line, helping you to confirm that you've drawn the line correctly. After plotting the intercepts, you can choose any value for \( x \), substitute it into the equation, and solve for \( y \) to find this third point.

In our example, for an x-value of 1, plug it into \( y = \frac{2}{3}x - 2 \) to get \( y = -\frac{4}{3} \). This gives a checkpoint of (1, -\frac{4}{3}), which you can plot on the graph to ensure your line is straight and accurate. Having this additional point helps provide a clear path for your line and reduces the likelihood of drawing errors. It's like having a friend point out that you're on the right track in a discussion - that extra reassurance is always helpful.