Certainly! To find the equation of a line that passes through the point [tex]\((5, 1)\)[/tex] and has a slope of [tex]\(\frac{1}{2}\)[/tex], we can use the point-slope form of a linear equation. The point-slope form is given by:
[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.
1. Identify the given point and slope:
- The point is [tex]\((5, 1)\)[/tex], so [tex]\(x_1 = 5\)[/tex] and [tex]\(y_1 = 1\)[/tex].
- The slope [tex]\(m\)[/tex] is [tex]\(\frac{1}{2}\)[/tex].
2. Substitute the point [tex]\((5, 1)\)[/tex] and the slope [tex]\(\frac{1}{2}\)[/tex] into the point-slope form equation:
[tex]\[ y - 1 = \frac{1}{2}(x - 5). \][/tex]
We can now verify the given choices to see which one matches this equation:
- Choice 1: [tex]\( y - 5 = \frac{1}{2}(x - 1) \)[/tex]
- This equation does not match our derived equation.
- Choice 2: [tex]\( y - \frac{1}{2} = 5(x - 1) \)[/tex]
- This equation does not match our derived equation.
- Choice 3: [tex]\( y - 1 = \frac{1}{2}(x - 5) \)[/tex]
- This equation exactly matches our derived equation.
- Choice 4: [tex]\( y - 1 = 5\left(x - \frac{1}{2}\right) \)[/tex]
- This equation does not match our derived equation.
Therefore, the correct equation is:
[tex]\[ y - 1 = \frac{1}{2}(x - 5). \][/tex]
So, the correct choice is:
[tex]\[ y - 1 = \frac{1}{2}(x - 5). \][/tex]
