To graph a line with a slope of [tex]\(-\frac{3}{4}\)[/tex] that passes through the point [tex]\((2, 3)\)[/tex], we will use the point-slope form of a linear equation. Here are the detailed steps:
1. Identify the point-slope form of the equation:
The point-slope form of a linear equation is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
where [tex]\((x_1, y_1)\)[/tex] is a point on the line and [tex]\(m\)[/tex] is the slope of the line.
2. Substitute the given point and slope into the equation:
Given:
[tex]\[
(x_1, y_1) = (2, 3)
\][/tex]
and
[tex]\[
m = -\frac{3}{4}
\][/tex]
Substitute these values into the point-slope form:
[tex]\[
y - 3 = -\frac{3}{4}(x - 2)
\][/tex]
3. Simplify the equation to the slope-intercept form (optional but easier for graphing):
Distribute the slope [tex]\(-\frac{3}{4}\)[/tex] through the parenthesis:
[tex]\[
y - 3 = -\frac{3}{4}x + \frac{3}{2}
\][/tex]
Add 3 to both sides to solve for [tex]\(y\)[/tex]:
[tex]\[
y = -\frac{3}{4}x + \frac{3}{2} + 3
\][/tex]
Convert 3 to a fraction with a common denominator:
[tex]\[
y = -\frac{3}{4}x + \frac{3}{2} + \frac{6}{2}
\][/tex]
Combine the fractions:
[tex]\[
y = -\frac{3}{4}x + \frac{9}{2}
\][/tex]
4. Write the equation in slope-intercept form:
The slope-intercept form is:
[tex]\[
y = mx + b
\][/tex]
So our equation is:
[tex]\[
y = -\frac{3}{4}x + \frac{9}{2}
\][/tex]
5. Plot the point [tex]\((2, 3)\)[/tex] on the graph:
Mark the point [tex]\((2, 3)\)[/tex] on the coordinate plane.
6. Use the slope to find another point:
Starting from [tex]\((2, 3)\)[/tex]:
- The slope [tex]\(-\frac{3}{4}\)[/tex] means that for every 4 units you move to the right along the x-axis, you move 3 units down along the y-axis.
- From [tex]\((2, 3)\)[/tex], move 4 units to the right to [tex]\(x = 6\)[/tex] and 3 units down to [tex]\(y = 0\)[/tex].
The second point is [tex]\((6, 0)\)[/tex].
7. Draw the line through the points:
- Plot the point [tex]\((6, 0)\)[/tex].
- Draw a straight line passing through both points [tex]\((2, 3)\)[/tex] and [tex]\((6, 0)\)[/tex].
8. Complete the graph:
Include labels for the axes and a title for the graph. You can also extend the line across the graph for better visualization.
Once you follow these steps, you'll have successfully graphed the line with a slope of [tex]\(-\frac{3}{4}\)[/tex] passing through the point [tex]\((2, 3)\)[/tex].
