Question

Graph the equation: \(3 x-2 y=6\)

Short answer

Plot y-intercept (0,-3), use slope 3/2 to find (2,0), draw line.

Step-by-step solution

- Write the equation in slope-intercept form

The given equation is \(3x - 2y = 6\). The slope-intercept form is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. First, solve for \(y\). Subtract \(3x\) from both sides: \(-2y = -3x + 6\). Then, divide by \(-2\) to isolate \(y\): \(y = \frac{3}{2}x - 3\).

- Identify the slope and y-intercept from the equation

From the equation \(y = \frac{3}{2}x - 3\), the slope \(m\) is \(\frac{3}{2}\) and the y-intercept \(b\) is \(-3\).

- Plot the y-intercept on the graph

To plot the y-intercept, locate the point \( (0, -3) \) on the y-axis and place a point there.

- Use the slope to find another point

The slope \( \frac{3}{2} \) means rise over run, or rise 3 units and run 2 units. Starting from \( (0, -3) \), move up 3 units to \(0 + 3 = 3\) and to the right 2 units to \(0 + 2 = 2\). Plot the point \( (2, 0) \).

- Draw the line

Draw a straight line through the points \( (0, -3) \) and \( (2, 0) \). This is the graph of the equation \( 3x - 2y = 6 \).

Key concepts

slope-intercept form

The slope-intercept form is an incredibly useful way to express linear equations. It lets you quickly identify the slope and the y-intercept of the line. The general formula is: \[y = mx + b\]where

• \(m\) is the slope of the line, which describes how steep the line is.
• \(b\) is the y-intercept, which is the point where the line crosses the y-axis.

In our exercise, we start with the standard equation \(3x - 2y = 6\) and transform it into the slope-intercept form. By isolating \(y\), we get the equation \(y = \frac{3}{2}x - 3\). This format makes it simple to read off the slope and y-intercept.

finding slope

The slope \(m\) describes how steep a line is on a graph. You can think of it as 'rise over run', which tells you how many units the line goes up (rise) for a certain number of units it goes across (run). For the equation \(y = \frac{3}{2}x - 3\), the slope is \(\frac{3}{2}\), or 1.5.
This means for every 2 units you move to the right on the x-axis, the line rises 3 units on the y-axis. When you see the slope written as a fraction, it helps you find other points on the line starting from the y-intercept.

y-intercept

The y-intercept is the point where the line crosses the y-axis. For our equation \(y = \frac{3}{2}x - 3\), the y-intercept is -3. This tells us that when \(x = 0\), \(y = -3\).
To plot this point, locate where the y-value is -3 on the y-axis, and place a point there. This serves as a starting reference point for graphing the entire line.

point plotting

To graph a line, after identifying the y-intercept and slope, you should plot multiple points. Starting from the y-intercept \((0, -3)\), use the slope to find other points.
Our slope is \(\frac{3}{2}\). This means rise 3 units and run 2 units.
Beginning at \((0, -3)\), we move up 3 units (a 'rise' to y = 0) and 2 units to the right (a 'run' to x = 2), landing at the point \((2, 0)\). Plot this point.

graphing lines

Now that we have our points \((0, -3)\) and \((2, 0)\) plotted, we can draw our line. Connect these points with a straight line extending in both directions. This line represents all the solutions to our original equation \(3x - 2y = 6\).
Remember to use a ruler for accuracy. Extending the line ensures that we can see the behavior of the equation over a range of values.