Let's determine the equation of the line given the slope ([tex]\(m\)[/tex]) and the y-intercept ([tex]\(b\)[/tex]) provided. The slope-intercept form of a line's equation is given by:
[tex]\[ y = mx + b \][/tex]
Here we are given:
- The slope ([tex]\(m\)[/tex]) is [tex]\(\frac{2}{3}\)[/tex].
- The y-intercept ([tex]\(b\)[/tex]) is [tex]\(-2\)[/tex].
Substituting these values into the slope-intercept formula, we get:
[tex]\[ y = \frac{2}{3} x + (-2) \][/tex]
This simplifies to:
[tex]\[ y = \frac{2}{3} x - 2 \][/tex]
This shows the equation of the line with a slope of [tex]\(\frac{2}{3}\)[/tex] and a y-intercept at [tex]\((0, -2)\)[/tex].
To find the correct answer among the given options, we compare our derived equation with the options:
A. [tex]\( y = -2x - \frac{2}{3} \)[/tex]
B. [tex]\( y = -\frac{3}{2} x - 2 \)[/tex]
C. [tex]\( y = -\frac{2}{3} x + \frac{2}{3} \)[/tex]
D. [tex]\( y = \frac{2}{3} x - 2 \)[/tex]
The only equation that matches [tex]\( y = \frac{2}{3} x - 2 \)[/tex] is:
D. [tex]\( y = \frac{2}{3} x - 2 \)[/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{D} \][/tex]
