Of course, let's go through the problem step by step to find which equation represents a line that passes through the point [tex]\((5, 1)\)[/tex] and has a slope of [tex]\(\frac{1}{2}\)[/tex].
1. Identify Point-Slope Form:
The point-slope form of a linear equation of a line with slope [tex]\(m\)[/tex] that passes through the point [tex]\((x_1, y_1)\)[/tex] is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
2. Substitute Given Values:
Here, the given slope [tex]\(m\)[/tex] is [tex]\(\frac{1}{2}\)[/tex], and the point [tex]\((x_1, y_1)\)[/tex] is [tex]\((5, 1)\)[/tex]. Substituting these into the point-slope form equation:
[tex]\[
y - 1 = \frac{1}{2}(x - 5)
\][/tex]
3. Compare with Given Choices:
Now we compare our derived equation [tex]\(y - 1 = \frac{1}{2}(x - 5)\)[/tex] with the given choices:
[tex]\[
\begin{array}{l}
y-5=\frac{1}{2}(x-1) \\
y-\frac{1}{2}=5(x-1) \\
y-1=\frac{1}{2}(x-5) \\
y-1=5\left(x-\frac{1}{2}\right)
\end{array}
\][/tex]
Clearly, the equation [tex]\(y - 1 = \frac{1}{2}(x - 5)\)[/tex] matches exactly with the given choice [tex]\(y-1=\frac{1}{2}(x-5)\)[/tex].
Therefore, the correct equation is:
[tex]\[
\boxed{y - 1 = \frac{1}{2}(x - 5)}
\][/tex]
Among the choices, this corresponds to the third option. Thus, the correct answer is indeed:
[tex]\[
\boxed{3}
\][/tex]
