Question

Explain how to graph \(2 x-3 y=6\) using slope-intercept form.

Short answer

First, convert the equation into slope-intercept form to get \(y = \frac{2}{3}x - 2\). The slope is \frac{2}{3} and the y-intercept is -2. First, plot the point (0, -2). Then, from that point, move 3 units to the right and 2 units up to plot the second point (3, 0). Finally, draw a line through these points to obtain the graph of the given equation.

Step-by-step solution

Convert Into Slope-Intercept Form

Rewrite the given equation \(2x - 3y = 6\) into the slope-intercept form. First, isolate \(y\) on one side. By doing this, we get: -3y = -2x + 6. Then, divide each term by -3, resulting in \(y = \frac{2}{3}x - 2\).

Identify the Slope and Y-Intercept

In the slope-intercept form \(y = mx + b\), \(m\) is the slope and \(b\) is the y-intercept. Therefore, comparing \(y = \frac{2}{3}x - 2\) to \(y = mx + b\), the slope \(m\) is \frac{2}{3}, and the y-intercept \(b\) is -2.

Plot the Y-Intercept

The y-intercept is the point where the line crosses the y-axis. In this case, it's at -2. Put a dot at coordinates (0, -2) on the graph.

Use the Slope to Plot Another Point

Once the y-intercept is plotted, the slope could be used to plot another point on the graph. With a slope of \frac{2}{3}, for every 3 units moving right (positive direction) along the x-axis from the y-intercept, move 2 units up (positive direction) along the y-axis. This will give another point (3, 0) to mark on the graph.

Draw the Line

Once you have at least two points, draw a line through them extending in both positive and negative direction. This line is the graph of the equation \(2x - 3y = 6\).

Key concepts

Slope-Intercept Form

Understanding the slope-intercept form is crucial when it comes to graphing linear equations. 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. The slope indicates the rise over the run, i.e., how much the line ascends (or descends) vertically for each horizontal step to the right. In contrast, the y-intercept is the exact point where the line crosses the y-axis.

To graph a linear equation effectively, start by rearranging the equation into this friendly form. For the equation \(2x - 3y = 6\), when you isolate \(y\) and rewrite it, the equation takes the form \(y = \frac{2}{3}x - 2\). Here, the slope is \(\frac{2}{3}\) and the y-intercept is \(-2\). With this formula, you're equipped to create a precise visual representation of the line on a graph.

Plotting Y-Intercept

The y-intercept serves as your starting point on the graph. It's essentially the launching pad from which you begin plotting your line. To plot the y-intercept of the equation \(y = \frac{2}{3}x - 2\), you look for the value of \(b\), which is \(-2\) in this case. Go to the y-axis on your graph, find the point that corresponds to \(-2\), and mark it.

This point, (0, -2), is where your line will cross the y-axis, and it's crucial because no matter how the rest of the graph looks, this point will always be on your line. You can consider it a guaranteed member of the set of solutions to the equation.

Using Slope to Plot Points

Once the y-intercept is marked, the slope will dictate where to plot the next points. The slope is a ratio, often represented as a fraction \(\frac{rise}{run}\). For the equation \(y = \frac{2}{3}x - 2\), the slope is \(\frac{2}{3}\). This means for every 3 units you move horizontally to the right from your y-intercept, you move 2 units vertically upward.

If you begin at the y-intercept (0, -2), move 3 units to the right, you'll be at x-coordinate 3. From here, ascend 2 units to locate the y-coordinate at point (3, 0). Now, with two points plotted (the y-intercept and this new point), you have enough information to draw a straight line extending through these points in both directions. This line is the graphical representation of your linear equation. Visualizing the slope this way makes it easier for anyone to understand and apply the concept.