Let’s determine the equation of a line in the slope-intercept form given a slope and y-intercept.
1. Understanding Slope-Intercept Form:
The slope-intercept form of a line is given by:
[tex]\[
y = mx + b
\][/tex]
where [tex]\( m \)[/tex] is the slope of the line and [tex]\( b \)[/tex] is the y-intercept.
2. Given Values:
- Slope ([tex]\( m \)[/tex]) = [tex]\(\frac{2}{3}\)[/tex]
- Y-intercept ([tex]\( b \)[/tex]) = -2
3. Substituting the Given Values:
By substituting the given values into the slope-intercept form:
[tex]\[
y = \frac{2}{3}x - 2
\][/tex]
Hence, the equation of the line with the given slope and y-intercept is:
[tex]\[
y = \frac{2}{3} x - 2
\][/tex]
4. Comparing with Given Options:
Now, let's match this equation with the given options:
- Option A: [tex]\( y = -\frac{3}{2} x - 2 \)[/tex]
- Option B: [tex]\( y = -\frac{2}{3} x + \frac{2}{3} \)[/tex]
- Option C: [tex]\( y = -2 x - \frac{2}{3} \)[/tex]
- Option D: [tex]\( y = \frac{2}{3} x - 2 \)[/tex]
Clearly, the equation we derived ([tex]\( y = \frac{2}{3} x - 2 \)[/tex]) matches Option D.
Therefore, the correct option is:
[tex]\[
\boxed{D}
\][/tex]
