To determine the graph of the equation [tex]\(v - 1 = \frac{2}{3}(x - 3)\)[/tex], we first need to rewrite it in the slope-intercept form, [tex]\(v = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept.
Here are the detailed steps to transform and understand the equation:
1. Start with the given equation:
[tex]\[
v - 1 = \frac{2}{3}(x - 3)
\][/tex]
2. Distribute [tex]\(\frac{2}{3}\)[/tex] on the right side:
[tex]\[
v - 1 = \frac{2}{3}x - \frac{2}{3} \cdot 3
\][/tex]
Simplifying the right side:
[tex]\[
v - 1 = \frac{2}{3}x - 2
\][/tex]
3. Add 1 to both sides to isolate [tex]\(v\)[/tex]:
[tex]\[
v = \frac{2}{3}x - 2 + 1
\][/tex]
Simplifying:
[tex]\[
v = \frac{2}{3}x - 1
\][/tex]
Now, we have the equation [tex]\(v = \frac{2}{3}x - 1\)[/tex] in slope-intercept form [tex]\(v = mx + b\)[/tex].
From this form, we can identify the following properties of the graph:
- The slope ([tex]\(m\)[/tex]) is [tex]\(\frac{2}{3}\)[/tex]. This indicates that for every 3 units increase in [tex]\(x\)[/tex], [tex]\(v\)[/tex] increases by 2 units.
- The y-intercept ([tex]\(b\)[/tex]) is [tex]\(-1\)[/tex]. This is the point where the line crosses the [tex]\(v\)[/tex]-axis.
### To Plot the Graph:
1. Start with the y-intercept:
- Plot the point [tex]\((0, -1)\)[/tex] on the graph. This is where the line will cross the [tex]\(v\)[/tex]-axis.
2. Use the slope to find another point:
- Starting from the y-intercept point [tex]\((0, -1)\)[/tex], use the slope [tex]\(\frac{2}{3}\)[/tex]:
- Move 3 units to the right (positive direction on the [tex]\(x\)[/tex]-axis)
- Move 2 units up (positive direction on the [tex]\(v\)[/tex]-axis)
- This gives you the point [tex]\((3, 1)\)[/tex].
3. Draw the line:
- Connect the points [tex]\((0, -1)\)[/tex] and [tex]\((3, 1)\)[/tex] with a straight line. Extend this line in both directions.
### Summary:
The graph of the equation [tex]\(v - 1 = \frac{2}{3}(x - 3)\)[/tex] is a straight line with a slope of [tex]\(\frac{2}{3}\)[/tex] and a y-intercept of [tex]\(-1\)[/tex]. The line rises 2 units for every 3 units of horizontal movement to the right.
