To determine the slope of the given line equation [tex]\( y - 3 = -\frac{1}{2}(x - 2) \)[/tex], we need to rewrite it in the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] represents the slope.
Let's simplify the given equation step-by-step:
1. Start with the given equation:
[tex]\[
y - 3 = -\frac{1}{2}(x - 2)
\][/tex]
2. Distribute the [tex]\(-\frac{1}{2}\)[/tex] across the terms inside the parentheses:
[tex]\[
y - 3 = -\frac{1}{2} \cdot x + \left(-\frac{1}{2} \cdot -2 \right)
\][/tex]
Simplify the multiplication inside the parentheses:
[tex]\[
y - 3 = -\frac{1}{2}x + 1
\][/tex]
3. Add 3 to both sides of the equation to isolate [tex]\( y \)[/tex]:
[tex]\[
y = -\frac{1}{2}x + 1 + 3
\][/tex]
4. Combine like terms on the right-hand side:
[tex]\[
y = -\frac{1}{2} x + 4
\][/tex]
Now that the equation is in the form [tex]\( y = mx + b \)[/tex], we can see that the coefficient of [tex]\( x \)[/tex] (the term [tex]\( m \)[/tex]) represents the slope of the line.
So, the slope [tex]\( m \)[/tex] is:
[tex]\[
m = -\frac{1}{2}
\][/tex]
Thus, the slope of the line with the equation [tex]\( y - 3 = -\frac{1}{2}(x - 2) \)[/tex] is [tex]\( -\frac{1}{2} \)[/tex]. The correct answer is not included in the given choices.
Please check the problem conditions, as the correct slope [tex]\( -\frac{1}{2} \)[/tex] is not listed as [tex]\(-2\)[/tex] or [tex]\(\frac{1}{2}\)[/tex].
