To graph the equation [tex]\( x + 3y = 6 \)[/tex], we'll go through the following steps:
### Step 1: Express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]
First, solve the equation for [tex]\( y \)[/tex]:
[tex]\[ x + 3y = 6 \][/tex]
Subtract [tex]\( x \)[/tex] from both sides:
[tex]\[ 3y = 6 - x \][/tex]
Divide both sides by 3:
[tex]\[ y = \frac{6 - x}{3} \][/tex]
### Step 2: Identify the y-intercept and x-intercept
The y-intercept occurs where [tex]\( x = 0 \)[/tex]:
[tex]\[ y = \frac{6 - 0}{3} = 2 \][/tex]
So, the y-intercept is [tex]\( (0, 2) \)[/tex].
The x-intercept occurs where [tex]\( y = 0 \)[/tex]:
[tex]\[ x + 3(0) = 6 \][/tex]
[tex]\[ x = 6 \][/tex]
So, the x-intercept is [tex]\( (6, 0) \)[/tex].
### Step 3: Choose additional points (optional)
To get a more accurate graph, we can choose additional values of [tex]\( x \)[/tex] and find the corresponding [tex]\( y \)[/tex]-values. For example:
- For [tex]\( x = -3 \)[/tex]:
[tex]\[ y = \frac{6 - (-3)}{3} = \frac{6 + 3}{3} = 3 \][/tex]
So, the point is [tex]\( (-3, 3) \)[/tex].
- For [tex]\( x = 3 \)[/tex]:
[tex]\[ y = \frac{6 - 3}{3} = 1 \][/tex]
So, the point is [tex]\( (3, 1) \)[/tex].
### Step 4: Plot the points on the coordinate plane
Using the points we found:
- [tex]\( (0, 2) \)[/tex]
- [tex]\( (6, 0) \)[/tex]
- [tex]\( (-3, 3) \)[/tex]
- [tex]\( (3, 1) \)[/tex]
### Step 5: Draw the graph
1. Draw the x-axis and y-axis.
2. Plot the points [tex]\( (0, 2) \)[/tex], [tex]\( (6, 0) \)[/tex], [tex]\( (-3, 3) \)[/tex], and [tex]\( (3, 1) \)[/tex] on the graph.
3. Draw a straight line through these points, extending in both directions.
Your graph should look like a straight line cutting the y-axis at [tex]\( (0, 2) \)[/tex] and the x-axis at [tex]\( (6, 0) \)[/tex], confirming the slope and intercepts we calculated.
Here is a representation of how the graph should look:
```
y
^
| .
| . |
| . |
|.------------|
-10 0 6 x
```
The line will extend through these points, forming a straight line within the coordinate plane.
