To determine which graph represents a line with a slope of [tex]\(-\frac{2}{3}\)[/tex] and a [tex]\(y\)[/tex]-intercept equal to that of the line [tex]\(y = \frac{2}{3}x - 2\)[/tex], we will follow these steps:
1. Identify the [tex]\(y\)[/tex]-intercept of the given line:
The equation of the line is given as [tex]\(y = \frac{2}{3}x - 2\)[/tex]. In a linear equation of the form [tex]\(y = mx + b\)[/tex], the term [tex]\(b\)[/tex] represents the [tex]\(y\)[/tex]-intercept. Therefore, from the equation [tex]\(y = \frac{2}{3}x - 2\)[/tex], the [tex]\(y\)[/tex]-intercept is [tex]\(-2\)[/tex].
2. Identify the specified slope:
The problem specifies that the slope of the required line is [tex]\(-\frac{2}{3}\)[/tex].
3. Write the equation of the required line:
We know the slope and the [tex]\(y\)[/tex]-intercept. The general form of the equation of a line is [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the [tex]\(y\)[/tex]-intercept. Given that the slope ([tex]\(m\)[/tex]) is [tex]\(-\frac{2}{3}\)[/tex] and the [tex]\(y\)[/tex]-intercept ([tex]\(b\)[/tex]) is [tex]\(-2\)[/tex], we can write the equation of the line as:
[tex]\[
y = -\frac{2}{3}x - 2
\][/tex]
4. Draw the graph:
To graph this line, start with the [tex]\(y\)[/tex]-intercept.
- Plot the point where the line crosses the [tex]\(y\)[/tex]-axis, which is at [tex]\(-2\)[/tex] on the [tex]\(y\)[/tex]-axis [tex]\((0, -2)\)[/tex].
- Use the slope to determine another point on the line. The slope of [tex]\(-\frac{2}{3}\)[/tex] means that for every 3 units you move to the right (positive direction on the [tex]\(x\)[/tex]-axis), you move 2 units down (negative direction on the [tex]\(y\)[/tex]-axis).
Starting from the [tex]\(y\)[/tex]-intercept [tex]\((0, -2)\)[/tex]:
- Move 3 units to the right: this brings you to [tex]\((3, -2)\)[/tex].
- Then move 2 units down: this gets you to the point [tex]\((3, -4)\)[/tex].
By plotting the points [tex]\((0, -2)\)[/tex] and [tex]\((3, -4)\)[/tex] and drawing a straight line through these points, you will have the graph of the line [tex]\(y = -\frac{2}{3}x - 2\)[/tex].
Therefore, the graph that depicts this line accurately will have a slope of [tex]\(-\frac{2}{3}\)[/tex] and will cross the [tex]\(y\)[/tex]-axis at [tex]\(-2\)[/tex].
