To find the slope of the line given by the equation [tex]\( y - 3 = -\frac{1}{2}(x - 2) \)[/tex], let's write this equation in the slope-intercept form, [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] represents the slope.
The given equation is [tex]\( y - 3 = -\frac{1}{2}(x - 2) \)[/tex].
1. Distribute the slope on the right-hand side:
[tex]\( y - 3 = -\frac{1}{2} \cdot x - \frac{1}{2} \cdot (-2) \)[/tex]
Simplify inside the parentheses:
[tex]\( y - 3 = -\frac{1}{2}x + 1 \)[/tex]
2. Isolate [tex]\( y \)[/tex] by adding 3 to both sides:
[tex]\( y = -\frac{1}{2}x + 1 + 3 \)[/tex]
3. Combine like terms on the right-hand side:
[tex]\( y = -\frac{1}{2}x + 4 \)[/tex]
Now the equation is in slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m = -\frac{1}{2} \)[/tex] and [tex]\( b = 4 \)[/tex].
Therefore, the slope of the line is [tex]\( -\frac{1}{2} \)[/tex].
So, the correct answer is:
[tex]\( -\frac{1}{2} \)[/tex].
