Sure! Let's graph the linear equation [tex]\( y = -x + 3 \)[/tex]. We'll go through this step-by-step:
### Step 1: Identify the slope and y-intercept
The equation [tex]\( y = -x + 3 \)[/tex] is in slope-intercept form [tex]\( y = mx + b \)[/tex], where:
- [tex]\( m \)[/tex] is the slope
- [tex]\( b \)[/tex] is the y-intercept
From the equation [tex]\( y = -x + 3 \)[/tex]:
- The slope [tex]\( m \)[/tex] is [tex]\(-1\)[/tex].
- The y-intercept [tex]\( b \)[/tex] is [tex]\( 3 \)[/tex].
### Step 2: Plot the y-intercept
The y-intercept is the point where the line crosses the y-axis. This occurs when [tex]\( x = 0 \)[/tex]:
[tex]\[ y = -0 + 3 = 3 \][/tex]
So, the y-intercept is at the point [tex]\( (0, 3) \)[/tex].
### Step 3: Use the slope to find another point on the line
The slope [tex]\( m = -1 \)[/tex] tells us that for every 1 unit increase in [tex]\( x \)[/tex], the value of [tex]\( y \)[/tex] decreases by 1 unit.
Starting from the y-intercept [tex]\( (0, 3) \)[/tex]:
- If [tex]\( x \)[/tex] increases by 1 unit: [tex]\( x = 1 \)[/tex]
- [tex]\( y \)[/tex] will decrease by 1 unit: [tex]\( y = 3 - 1 \)[/tex]
This gives us the point:
[tex]\[ (1, 3 - 1) = (1, 2) \][/tex]
You can also find more points by continuing this pattern. For example:
- If [tex]\( x \)[/tex] increases to [tex]\( 2 \)[/tex]: [tex]\( x = 2 \)[/tex]
- [tex]\( y \)[/tex] will decrease by 2 units from the y-intercept: [tex]\( y = 3 - 2 \)[/tex]
This gives us another point:
[tex]\[ (2, 3 - 2) = (2, 1) \][/tex]
If [tex]\( x \)[/tex] increases to 3, then:
[tex]\[ y = 3 - 3 = 0 \][/tex]
This gives us the point:
[tex]\[ (3, 0) \][/tex]
And so on. Similarly, if [tex]\( x \)[/tex] decreases (goes negative), then [tex]\( y \)[/tex] will increase because [tex]\( -(-1)x = x \)[/tex]:
If [tex]\( x = -1 \)[/tex]:
[tex]\[ y = -(-1) + 3 = 1 + 3 = 4 \][/tex]
So the point is [tex]\( (-1, 4) \)[/tex].
### Step 4: Plot the points and draw the line
Use graph paper or plotting software to plot the following points:
- [tex]\( (0, 3) \)[/tex]
- [tex]\( (1, 2) \)[/tex]
- [tex]\( (2, 1) \)[/tex]
- [tex]\( (3, 0) \)[/tex]
- [tex]\( (-1, 4) \)[/tex]
Once you have these points plotted, draw a straight line through them. Every point along this line satisfies the equation [tex]\( y = -x + 3 \)[/tex].
### Step 5: Verify the line
Make sure that the line diminishes by 1 unit in [tex]\( y \)[/tex] for every unit increase in [tex]\( x \)[/tex], and vice versa.
The resulting graph should look like this:
```
y
|
4| .
3| .
2| .
1| .
----|--------|--------|---- x
-2 -1 0 1 2 3
```
The straight line through these points is the graph of the linear equation [tex]\( y = -x + 3 \)[/tex].
