To graph the linear equation [tex]\(3x - 2y = 6\)[/tex], follow these steps to derive its graphical representation:
### Step 1: Solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]
First, we need to rewrite the equation in the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
Given:
[tex]\[ 3x - 2y = 6 \][/tex]
Isolate [tex]\( y \)[/tex] by following these steps:
1. Subtract [tex]\( 3x \)[/tex] from both sides:
[tex]\[ -2y = -3x + 6 \][/tex]
2. Divide both sides by [tex]\(-2\)[/tex]:
[tex]\[ y = \frac{-3x + 6}{-2} \][/tex]
Simplify the fraction:
[tex]\[ y = \frac{3}{2}x - 3 \][/tex]
So, the equation in slope-intercept form is:
[tex]\[ y = \frac{3}{2}x - 3 \][/tex]
### Step 2: Identify the slope and y-intercept
From the equation [tex]\( y = \frac{3}{2}x - 3 \)[/tex]:
- The slope ([tex]\( m \)[/tex]) is [tex]\( \frac{3}{2} \)[/tex].
- The y-intercept ([tex]\( b \)[/tex]) is [tex]\(-3\)[/tex], which means the line crosses the y-axis at [tex]\( (0, -3) \)[/tex].
### Step 3: Determine key points
To graph the line accurately, determine additional points besides the y-intercept.
1. The y-intercept is [tex]\( (0, -3) \)[/tex].
2. Find the x-intercept, where [tex]\( y = 0 \)[/tex]:
Substitute [tex]\( y = 0 \)[/tex] in the original equation [tex]\( 3x - 2(0) = 6 \)[/tex]:
[tex]\[ 3x = 6 \][/tex]
[tex]\[ x = 2 \][/tex]
So, the x-intercept is [tex]\( (2, 0) \)[/tex].
### Step 4: Plot the points and draw the line
1. Plot the y-intercept [tex]\( (0, -3) \)[/tex] on the graph.
2. Plot the x-intercept [tex]\( (2, 0) \)[/tex] on the graph.
### Step 5: Draw the line
Draw a straight line through these two points, extending it in both directions.
### Visual Representation
To help visualize, here is a sketch of the graph:
```
y
|
4|
3|
2|
1|
0|___________________________________________ x
-2 0 2
(-3) (2,0)
-1|
-2|
-3|
-4|
```
In this graph:
- The y-intercept is at [tex]\( (0, -3) \)[/tex].
- The x-intercept is at [tex]\( (2, 0) \)[/tex].
This line has a positive slope of [tex]\( \frac{3}{2} \)[/tex], indicating that as [tex]\( x \)[/tex] increases, [tex]\( y \)[/tex] increases by [tex]\( \frac{3}{2} \)[/tex] times the amount of the increase in [tex]\( x \)[/tex].
