To find the graph representing a line with a slope of [tex]\(-\frac{2}{3}\)[/tex] and the same [tex]\(y\)[/tex]-intercept as the line [tex]\(y = \frac{2}{3} x - 2\)[/tex], follow these steps:
1. Identify the [tex]\(y\)[/tex]-intercept of the given line:
The equation [tex]\(y = \frac{2}{3} x - 2\)[/tex] is in the slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the [tex]\(y\)[/tex]-intercept. For this equation, the [tex]\(y\)[/tex]-intercept [tex]\(b\)[/tex] is [tex]\(-2\)[/tex].
2. Define the slope:
We need to use a slope of [tex]\(-\frac{2}{3}\)[/tex] for our new line.
3. Write the equation of the new line:
Since we have a slope of [tex]\(-\frac{2}{3}\)[/tex] and a [tex]\(y\)[/tex]-intercept of [tex]\(-2\)[/tex], we can write the equation of the new line as:
[tex]\[
y = -\frac{2}{3} x - 2
\][/tex]
4. Graphing the line:
To graph this line, we will:
- Start at the [tex]\(y\)[/tex]-intercept [tex]\((0, -2)\)[/tex] on the graph.
- Use the slope [tex]\(-\frac{2}{3}\)[/tex] to find another point on the line. A slope of [tex]\(-\frac{2}{3}\)[/tex] means that for every 3 units we move horizontally to the right, we move 2 units down.
Therefore, starting from [tex]\((0, -2)\)[/tex]:
- Move 3 units to the right, we arrive at [tex]\(x = 3\)[/tex].
- Move 2 units down from [tex]\(y = -2\)[/tex], we end up at [tex]\(y = -4\)[/tex].
Thus, another point on the line is [tex]\((3, -4)\)[/tex].
5. Plot the points and draw the line:
- Plot the first point [tex]\((0, -2)\)[/tex].
- Plot the second point [tex]\((3, -4)\)[/tex].
- Draw a straight line through these two points. This line represents the equation [tex]\(y = -\frac{2}{3} x - 2\)[/tex].
The graph you obtain by following these steps is the one that correctly represents a line with a slope of [tex]\(-\frac{2}{3}\)[/tex] and a [tex]\(y\)[/tex]-intercept of [tex]\(-2\)[/tex].
