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Sagot :
Sure! Let's solve the problem step-by-step in order to graph the line given by the equation:
[tex]\[ y - 2 = -\frac{3}{4}(x - 1) \][/tex]
### Step 1: Identify the Slope and a Point on the Line
The given equation is in point-slope form:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Here, the slope [tex]\( m = -\frac{3}{4} \)[/tex] and the point [tex]\((x_1, y_1)\)[/tex] on the line is [tex]\((1, 2)\)[/tex].
### Step 2: Convert to Slope-Intercept Form
To graph the line more easily, let's convert the given equation to slope-intercept form [tex]\( y = mx + b \)[/tex].
Starting with:
[tex]\[ y - 2 = -\frac{3}{4}(x - 1) \][/tex]
Distribute the slope on the right-hand side:
[tex]\[ y - 2 = -\frac{3}{4}x + \frac{3}{4} \][/tex]
Add 2 to both sides to solve for [tex]\( y \)[/tex]:
[tex]\[ y = -\frac{3}{4}x + \frac{3}{4} + 2 \][/tex]
Combine constants on the right-hand side:
[tex]\[ y = -\frac{3}{4}x + 2.75 \][/tex]
So, the slope-intercept form is:
[tex]\[ y = -\frac{3}{4}x + 2.75 \][/tex]
Here, the slope [tex]\( m = -\frac{3}{4} \)[/tex] and the y-intercept [tex]\( b = 2.75 \)[/tex].
### Step 3: Plot the Line
Now we can plot the line using the information derived.
1. Plot the y-intercept (0, 2.75): This is where the line crosses the y-axis.
2. Use the slope to find another point:
- The slope [tex]\( m = -\frac{3}{4} \)[/tex] means that for every 4 units you move to the right on the x-axis, you move 3 units down on the y-axis.
- Starting from the y-intercept (0, 2.75), if we move 4 units to the right to [tex]\( x = 4 \)[/tex], then [tex]\( y = 2.75 - 3 = -0.25 \)[/tex].
- So another point is [tex]\( (4, -0.25) \)[/tex].
3. Draw the line:
- Plot the points [tex]\( (0, 2.75) \)[/tex] and [tex]\( (4, -0.25) \)[/tex].
- Draw a straight line through these points, extending in both directions.
### Step 4: Verify Points for Accuracy
For accuracy, it's helpful to calculate a few points on the line:
- Point calculations (using linear values within the typical plotting range):
| x | y |
|---------|----------------|
| -10 | 10.25 |
| -9.949 | 10.212 |
| -9.899 | 10.175 |
| -9.849 | 10.137 |
| -9.799 | 10.100 |
These points confirm that our calculations are correct, as they lie on the expected line.
### Conclusion
By plotting these points and drawing the line through them, you will graph the line described by the equation [tex]\( y - 2 = -\frac{3}{4}(x - 1) \)[/tex].
Here is the visual representation of the line:
1. Starting at the y-intercept (0, 2.75)
2. Using the slope to find additional points like (4, -0.25)
3. Drawing a continuous line through these points
This visual graph depicts the relationship defined by the linear equation given.
[tex]\[ y - 2 = -\frac{3}{4}(x - 1) \][/tex]
### Step 1: Identify the Slope and a Point on the Line
The given equation is in point-slope form:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Here, the slope [tex]\( m = -\frac{3}{4} \)[/tex] and the point [tex]\((x_1, y_1)\)[/tex] on the line is [tex]\((1, 2)\)[/tex].
### Step 2: Convert to Slope-Intercept Form
To graph the line more easily, let's convert the given equation to slope-intercept form [tex]\( y = mx + b \)[/tex].
Starting with:
[tex]\[ y - 2 = -\frac{3}{4}(x - 1) \][/tex]
Distribute the slope on the right-hand side:
[tex]\[ y - 2 = -\frac{3}{4}x + \frac{3}{4} \][/tex]
Add 2 to both sides to solve for [tex]\( y \)[/tex]:
[tex]\[ y = -\frac{3}{4}x + \frac{3}{4} + 2 \][/tex]
Combine constants on the right-hand side:
[tex]\[ y = -\frac{3}{4}x + 2.75 \][/tex]
So, the slope-intercept form is:
[tex]\[ y = -\frac{3}{4}x + 2.75 \][/tex]
Here, the slope [tex]\( m = -\frac{3}{4} \)[/tex] and the y-intercept [tex]\( b = 2.75 \)[/tex].
### Step 3: Plot the Line
Now we can plot the line using the information derived.
1. Plot the y-intercept (0, 2.75): This is where the line crosses the y-axis.
2. Use the slope to find another point:
- The slope [tex]\( m = -\frac{3}{4} \)[/tex] means that for every 4 units you move to the right on the x-axis, you move 3 units down on the y-axis.
- Starting from the y-intercept (0, 2.75), if we move 4 units to the right to [tex]\( x = 4 \)[/tex], then [tex]\( y = 2.75 - 3 = -0.25 \)[/tex].
- So another point is [tex]\( (4, -0.25) \)[/tex].
3. Draw the line:
- Plot the points [tex]\( (0, 2.75) \)[/tex] and [tex]\( (4, -0.25) \)[/tex].
- Draw a straight line through these points, extending in both directions.
### Step 4: Verify Points for Accuracy
For accuracy, it's helpful to calculate a few points on the line:
- Point calculations (using linear values within the typical plotting range):
| x | y |
|---------|----------------|
| -10 | 10.25 |
| -9.949 | 10.212 |
| -9.899 | 10.175 |
| -9.849 | 10.137 |
| -9.799 | 10.100 |
These points confirm that our calculations are correct, as they lie on the expected line.
### Conclusion
By plotting these points and drawing the line through them, you will graph the line described by the equation [tex]\( y - 2 = -\frac{3}{4}(x - 1) \)[/tex].
Here is the visual representation of the line:
1. Starting at the y-intercept (0, 2.75)
2. Using the slope to find additional points like (4, -0.25)
3. Drawing a continuous line through these points
This visual graph depicts the relationship defined by the linear equation given.
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