To determine which equation represents a line passing through the point [tex]\((5, 1)\)[/tex] with a slope of [tex]\(\frac{1}{2}\)[/tex], we should use the point-slope form of the equation of a line. The point-slope form is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where [tex]\((x_1, y_1)\)[/tex] is a point on the line and [tex]\(m\)[/tex] is the slope.
Given:
- the point [tex]\((5, 1)\)[/tex] means [tex]\(x_1 = 5\)[/tex] and [tex]\(y_1 = 1\)[/tex]
- the slope [tex]\(m = \frac{1}{2}\)[/tex]
We substitute these values into the point-slope form equation:
[tex]\[ y - 1 = \frac{1}{2}(x - 5) \][/tex]
Therefore, the equation of the line that passes through the point [tex]\((5, 1)\)[/tex] and has a slope of [tex]\(\frac{1}{2}\)[/tex] is:
[tex]\[ y - 1 = \frac{1}{2}(x - 5) \][/tex]
Thus, the correct option is:
[tex]\[ y - 1 = \frac{1}{2}(x - 5) \][/tex]
This matches with the third option:
[tex]\[ y - 1 = \frac{1}{2}(x - 5) \][/tex]
