To graph the linear equation [tex]\(3x - 2y = 6\)[/tex], we will follow these steps:
1. Rewrite the equation in slope-intercept form [tex]\(y = mx + b\)[/tex]:
- Start with the given equation: [tex]\(3x - 2y = 6\)[/tex].
- Solve for [tex]\(y\)[/tex] by isolating it on one side of the equation.
[tex]\[
3x - 2y = 6 \implies -2y = -3x + 6 \implies y = \frac{3}{2}x - 3
\][/tex]
This equation is now in the slope-intercept form [tex]\(y = mx + b\)[/tex], where the slope [tex]\(m\)[/tex] is [tex]\(\frac{3}{2}\)[/tex] and the y-intercept [tex]\(b\)[/tex] is [tex]\(-3\)[/tex].
2. Determine the key points to plot the line:
- Identify the y-intercept: This occurs when [tex]\(x = 0\)[/tex].
[tex]\[
y = \frac{3}{2}(0) - 3 = -3
\][/tex]
Thus, one point on the line is [tex]\((0, -3)\)[/tex].
- Choose another x-value to find a second point on the line. For simplicity, we can use [tex]\(x = 2\)[/tex]:
[tex]\[
y = \frac{3}{2}(2) - 3 = 3 - 3 = 0
\][/tex]
So, another point on the line is [tex]\((2, 0)\)[/tex].
3. Plot these points on the graph:
- Plot [tex]\((0, -3)\)[/tex] and [tex]\((2, 0)\)[/tex] on the coordinate system.
- Draw a straight line that passes through these points. This line represents the equation [tex]\(3x - 2y = 6\)[/tex].
4. Create a table of values for more points if required:
- It's often useful to have more points to ensure the accuracy of the graph. Here are some additional x-values and their corresponding y-values:
- For [tex]\(x = -2\)[/tex]:
[tex]\[
y = \frac{3}{2}(-2) - 3 = -3 - 3 = -6 \quad \text{thus, point } (-2, -6)
\][/tex]
- For [tex]\(x = 4\)[/tex]:
[tex]\[
y = \frac{3}{2}(4) - 3 = 6 - 3 = 3 \quad \text{thus, point } (4, 3)
\][/tex]
Using a broader range of x-values, here are some specific points from our list:
- [tex]\((-10, -18)\)[/tex]
- [tex]\((-5, -7.5)\)[/tex]
- [tex]\((0, -3)\)[/tex]
- [tex]\((5, 4.5)\)[/tex]
- [tex]\((10, 12)\)[/tex]
5. Graphically represent the line:
- Connect all the plotted points with a straight line.
6. Analyze the graph:
- The slope [tex]\(\frac{3}{2}\)[/tex] indicates that for every 2 units we move to the right, we move up by 3 units.
- The graph is a straight line that either extends infinitely in both directions, confirming it is a linear relationship.
Below is the approximate plot of these points:
```
y
|
| (10,12)
| /
| /
| /
| /
| / (4,3)
|/__________ x
(0,-3) \
(-5,-7.5)\
(-10,-18) \
```
Conclusion:
The line passing through the points [tex]\((0, -3)\)[/tex], [tex]\((2, 0)\)[/tex], [tex]\((-2, -6)\)[/tex], and others as calculated represents the graph of the equation [tex]\(3x - 2y = 6\)[/tex].
