To determine the equation of a line given its slope and [tex]\( y \)[/tex]-intercept, we use the slope-intercept form of a linear equation, which is:
[tex]\[ y = mx + b \][/tex]
where:
- [tex]\( m \)[/tex] is the slope of the line,
- [tex]\( b \)[/tex] is the [tex]\( y \)[/tex]-intercept.
### Step-by-Step Solution:
1. Identify the given values:
- The slope ([tex]\( m \)[/tex]) is [tex]\(\frac{3}{2}\)[/tex].
- The [tex]\( y \)[/tex]-intercept ([tex]\( b \)[/tex]) is [tex]\(-12\)[/tex].
2. Substitute these values into the slope-intercept form:
- The slope-intercept form equation is [tex]\( y = mx + b \)[/tex].
- Plug in the slope [tex]\( m = \frac{3}{2} \)[/tex] and [tex]\( y \)[/tex]-intercept [tex]\( b = -12 \)[/tex].
[tex]\[ y = \left(\frac{3}{2}\right)x + (-12) \][/tex]
[tex]\[ y = \left(\frac{3}{2}\right)x - 12 \][/tex]
3. Match this equation with the provided options:
- A. [tex]\( y = \frac{3}{2} x - 12 \)[/tex]
- B. [tex]\( y = \frac{3}{2} x + 12 \)[/tex]
- C. [tex]\( y = \frac{2}{3} x - 12 \)[/tex]
- D. [tex]\( y = -12 x - \frac{3}{2} \)[/tex]
The correct equation that matches our derived equation is:
[tex]\[ \boxed{y = \frac{3}{2} x - 12} \][/tex]
Thus, the correct answer is A.
