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], let's follow these steps in detail:
1. Identify the given line's slope and [tex]\(y\)[/tex]-intercept:
The equation of the given line is [tex]\( y = \frac{2}{3}x - 2 \)[/tex].
- The slope ([tex]\(m_1\)[/tex]) of this line is [tex]\(\frac{2}{3}\)[/tex].
- The [tex]\(y\)[/tex]-intercept ([tex]\(c_1\)[/tex]) of this line is [tex]\(-2\)[/tex], which is the constant term in the equation.
2. Determine the required line's characteristics:
We need to find a new line that has:
- A slope of [tex]\(-\frac{2}{3}\)[/tex].
- The same [tex]\(y\)[/tex]-intercept as the given line, which is [tex]\(-2\)[/tex].
3. Write the equation of the new line:
Using the slope-intercept form of a line, [tex]\(y = mx + c\)[/tex], where:
- [tex]\(m_2 = -\frac{2}{3}\)[/tex] is the slope of the new line.
- [tex]\(c_2 = -2\)[/tex] is the [tex]\(y\)[/tex]-intercept of the new line, same as the given line.
Thus, the equation of the new line is:
[tex]\[
y = -\frac{2}{3} x - 2
\][/tex]
4. Interpret the graphical representation:
A graph representing the line [tex]\( y = -\frac{2}{3}x - 2 \)[/tex] will have:
- A slope of [tex]\(-\frac{2}{3}\)[/tex], meaning it descends to the right.
- A [tex]\(y\)[/tex]-intercept of [tex]\(-2\)[/tex], where the line crosses the [tex]\(y\)[/tex]-axis at [tex]\((0, -2)\)[/tex].
Visually, you would look for a graph where the line passes through the point [tex]\((0, -2)\)[/tex] and as you move 3 units to the right along the [tex]\(x\)[/tex]-axis, the line descends by 2 units.
Based on the given result, the correct graph accurately represents a line with a slope of [tex]\(-\frac{2}{3}\)[/tex] and a [tex]\(y\)[/tex]-intercept of [tex]\(-2\)[/tex].
