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To graph the line with a slope of [tex]\(-\frac{3}{4}\)[/tex] that contains the point [tex]\((2, 3)\)[/tex], we can follow a systematic approach. Here's a detailed step-by-step solution:
1. Identify the Information Given:
- Slope ([tex]\(m\)[/tex]): [tex]\(-\frac{3}{4}\)[/tex]
- A point on the line: [tex]\((2, 3)\)[/tex]
2. Use the Point-Slope Form Equation of a Line:
The point-slope form of the equation of a line is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where [tex]\((x_1, y_1)\)[/tex] is the given point and [tex]\(m\)[/tex] is the slope.
Plugging in the values:
[tex]\[ y - 3 = -\frac{3}{4}(x - 2) \][/tex]
3. Simplify the Equation to Solve for the Y-Intercept:
To find the y-intercept ([tex]\(b\)[/tex]), we need to convert this to the slope-intercept form [tex]\(y = mx + b\)[/tex].
Expanding the equation:
[tex]\[ y - 3 = -\frac{3}{4}x + \frac{3}{2} \][/tex]
Add 3 to both sides:
[tex]\[ y = -\frac{3}{4}x + \frac{3}{2} + 3 \][/tex]
Combine the constants:
[tex]\[ y = -\frac{3}{4}x + \frac{3}{2} + \frac{6}{2} \][/tex]
[tex]\[ y = -\frac{3}{4}x + \frac{9}{2} \][/tex]
Therefore, the slope-intercept form of the equation is:
[tex]\[ y = -\frac{3}{4}x + 4.5 \][/tex]
Thus, the y-intercept [tex]\(b\)[/tex] is 4.5.
4. Identify Points on the Line:
We can calculate some points on the line to help in graphing.
For example, let's use [tex]\(x\)[/tex] values of [tex]\(-5, 0, 2, 5, 10\)[/tex] and find corresponding [tex]\(y\)[/tex] values.
- When [tex]\(x = -5\)[/tex]:
[tex]\[ y = -\frac{3}{4}(-5) + 4.5 = 3.75 + 4.5 = 8.25 \][/tex]
- When [tex]\(x = 0\)[/tex]:
[tex]\[ y = -\frac{3}{4}(0) + 4.5 = 4.5 \][/tex]
- When [tex]\(x = 2\)[/tex]:
[tex]\[ y = -\frac{3}{4}(2) + 4.5 = -1.5 + 4.5 = 3 \][/tex]
- When [tex]\(x = 5\)[/tex]:
[tex]\[ y = -\frac{3}{4}(5) + 4.5 = -3.75 + 4.5 = 0.75 \][/tex]
- When [tex]\(x = 10\)[/tex]:
[tex]\[ y = -\frac{3}{4}(10) + 4.5 = -7.5 + 4.5 = -3 \][/tex]
So, the corresponding points are:
[tex]\[ (x, y): (-5, 8.25), (0, 4.5), (2, 3), (5, 0.75), (10, -3) \][/tex]
5. Graph the Line:
- Begin by plotting the given point [tex]\((2, 3)\)[/tex] on the graph.
- Then, plot the other points calculated: [tex]\((-5, 8.25)\)[/tex], [tex]\((0, 4.5)\)[/tex], [tex]\((5, 0.75)\)[/tex], and [tex]\((10, -3)\)[/tex].
- Draw a straight line passing through all these points.
This method gives a detailed roadmap to correctly graph the line with the slope [tex]\(-\frac{3}{4}\)[/tex] passing through the point [tex]\((2, 3)\)[/tex].
1. Identify the Information Given:
- Slope ([tex]\(m\)[/tex]): [tex]\(-\frac{3}{4}\)[/tex]
- A point on the line: [tex]\((2, 3)\)[/tex]
2. Use the Point-Slope Form Equation of a Line:
The point-slope form of the equation of a line is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where [tex]\((x_1, y_1)\)[/tex] is the given point and [tex]\(m\)[/tex] is the slope.
Plugging in the values:
[tex]\[ y - 3 = -\frac{3}{4}(x - 2) \][/tex]
3. Simplify the Equation to Solve for the Y-Intercept:
To find the y-intercept ([tex]\(b\)[/tex]), we need to convert this to the slope-intercept form [tex]\(y = mx + b\)[/tex].
Expanding the equation:
[tex]\[ y - 3 = -\frac{3}{4}x + \frac{3}{2} \][/tex]
Add 3 to both sides:
[tex]\[ y = -\frac{3}{4}x + \frac{3}{2} + 3 \][/tex]
Combine the constants:
[tex]\[ y = -\frac{3}{4}x + \frac{3}{2} + \frac{6}{2} \][/tex]
[tex]\[ y = -\frac{3}{4}x + \frac{9}{2} \][/tex]
Therefore, the slope-intercept form of the equation is:
[tex]\[ y = -\frac{3}{4}x + 4.5 \][/tex]
Thus, the y-intercept [tex]\(b\)[/tex] is 4.5.
4. Identify Points on the Line:
We can calculate some points on the line to help in graphing.
For example, let's use [tex]\(x\)[/tex] values of [tex]\(-5, 0, 2, 5, 10\)[/tex] and find corresponding [tex]\(y\)[/tex] values.
- When [tex]\(x = -5\)[/tex]:
[tex]\[ y = -\frac{3}{4}(-5) + 4.5 = 3.75 + 4.5 = 8.25 \][/tex]
- When [tex]\(x = 0\)[/tex]:
[tex]\[ y = -\frac{3}{4}(0) + 4.5 = 4.5 \][/tex]
- When [tex]\(x = 2\)[/tex]:
[tex]\[ y = -\frac{3}{4}(2) + 4.5 = -1.5 + 4.5 = 3 \][/tex]
- When [tex]\(x = 5\)[/tex]:
[tex]\[ y = -\frac{3}{4}(5) + 4.5 = -3.75 + 4.5 = 0.75 \][/tex]
- When [tex]\(x = 10\)[/tex]:
[tex]\[ y = -\frac{3}{4}(10) + 4.5 = -7.5 + 4.5 = -3 \][/tex]
So, the corresponding points are:
[tex]\[ (x, y): (-5, 8.25), (0, 4.5), (2, 3), (5, 0.75), (10, -3) \][/tex]
5. Graph the Line:
- Begin by plotting the given point [tex]\((2, 3)\)[/tex] on the graph.
- Then, plot the other points calculated: [tex]\((-5, 8.25)\)[/tex], [tex]\((0, 4.5)\)[/tex], [tex]\((5, 0.75)\)[/tex], and [tex]\((10, -3)\)[/tex].
- Draw a straight line passing through all these points.
This method gives a detailed roadmap to correctly graph the line with the slope [tex]\(-\frac{3}{4}\)[/tex] passing through the point [tex]\((2, 3)\)[/tex].
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