Let's start by converting the given equation from point-slope form to slope-intercept form. The point-slope form of the equation given is:
[tex]\[ y - 3 = \frac{1}{2}(x - 1) \][/tex]
To convert this to the slope-intercept form \( y = mx + b \), we need to:
1. Distribute the slope on the right-hand side.
2. Isolate \( y \) on the left-hand side.
First, let's distribute \(\frac{1}{2}\) to both \(x\) and \(-1\):
[tex]\[ y - 3 = \frac{1}{2} x - \frac{1}{2} \][/tex]
Next, we need to isolate \( y \). To do this, we add 3 to both sides of the equation:
[tex]\[ y = \frac{1}{2} x - \frac{1}{2} + 3 \][/tex]
Now, let's combine the constants on the right-hand side:
[tex]\[ y = \frac{1}{2} x - \frac{1}{2} + \frac{6}{2} \][/tex]
Since \( 3 \) can be written as \( \frac{6}{2} \), we add the constants:
[tex]\[ y = \frac{1}{2} x + \frac{5}{2} \][/tex]
So, the slope-intercept form of the equation is:
[tex]\[ y = \frac{1}{2} x + \frac{5}{2} \][/tex]
Among the given choices, this matches the third option:
[tex]\[ y = \frac{1}{2} x + \frac{5}{2} \][/tex]
Therefore, the slope-intercept form of the equation for this line is:
[tex]\[ y = \frac{1}{2} x + \frac{5}{2} \][/tex]
So, the correct choice is:
[tex]\[ \boxed{y = \frac{1}{2}x + \frac{5}{2}} \][/tex]
