To graph the line with slope [tex]\(-\frac{1}{3}\)[/tex] that passes through the point [tex]\((5, 3)\)[/tex], follow these steps:
1. Determine the slope-intercept form of the line:
The slope-intercept form of a line is given by [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept.
Here, the slope [tex]\(m\)[/tex] is given as [tex]\(-\frac{1}{3}\)[/tex]. The equation can be written as:
[tex]\[
y = -\frac{1}{3}x + b
\][/tex]
2. Find the y-intercept ([tex]\(b\)[/tex]):
To find the y-intercept, use the given point [tex]\((5, 3)\)[/tex]. Substitute [tex]\(x = 5\)[/tex] and [tex]\(y = 3\)[/tex] into the equation [tex]\(y = -\frac{1}{3}x + b\)[/tex]:
[tex]\[
3 = -\frac{1}{3}(5) + b
\][/tex]
Simplify the equation:
[tex]\[
3 = -\frac{5}{3} + b
\][/tex]
To isolate [tex]\(b\)[/tex], add [tex]\(\frac{5}{3}\)[/tex] to both sides:
[tex]\[
3 + \frac{5}{3} = b
\][/tex]
Convert [tex]\(3\)[/tex] to a fraction with the same denominator:
[tex]\[
\frac{9}{3} + \frac{5}{3} = b
\][/tex]
Combine the fractions:
[tex]\[
b = \frac{14}{3}
\][/tex]
3. Write the equation of the line:
Now that we have the slope and the y-intercept, the equation of the line is:
[tex]\[
y = -\frac{1}{3}x + \frac{14}{3}
\][/tex]
4. Plot the line:
- Start by plotting the y-intercept on the graph. The y-intercept [tex]\(b = \frac{14}{3}\)[/tex] is approximately [tex]\(4.67\)[/tex].
- Plot the point [tex]\((0, \frac{14}{3})\)[/tex].
5. Use the slope to determine another point:
- From the y-intercept, use the slope to find another point on the line. The slope [tex]\(-\frac{1}{3}\)[/tex] means that for every 3 units you move horizontally to the right, you move 1 unit down.
- Starting from [tex]\((0, \frac{14}{3})\)[/tex], move 3 units right to [tex]\((3, \frac{14}{3} - 1)\)[/tex], which is [tex]\((3, \frac{14}{3} - \frac{3}{3})\)[/tex] or [tex]\((3, \frac{11}{3}) \approx (3, 3.67)\)[/tex].
- Alternatively, you can easily plot the given point [tex]\((5, 3)\)[/tex] as a check.
6. Draw the line:
- Connect the points [tex]\((0, \frac{14}{3})\)[/tex] and [tex]\((5, 3)\)[/tex] with a straight line extending in both directions.
By following the steps above, you should be able to accurately graph the line with slope [tex]\(-\frac{1}{3}\)[/tex] that passes through the point [tex]\((5, 3)\)[/tex].
