Answer :
Certainly! Let's find which equation represents a line passing through the point [tex]\((5, 1)\)[/tex] with a slope of [tex]\(\frac{1}{2}\)[/tex] using the point-slope form.
The point-slope form of the equation of a line is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Where:
- [tex]\((x_1, y_1)\)[/tex] is a point on the line.
- [tex]\(m\)[/tex] is the slope of the line.
In this case:
- The point [tex]\((x_1, y_1)\)[/tex] is [tex]\((5, 1)\)[/tex].
- The slope [tex]\(m\)[/tex] is [tex]\(\frac{1}{2}\)[/tex].
Substituting the given values into the point-slope form equation, we get:
[tex]\[ y - 1 = \frac{1}{2}(x - 5) \][/tex]
Now, let's compare this with the given options:
1. [tex]\( y - 5 = \frac{1}{2}(x - 1) \)[/tex]
2. [tex]\( y - \frac{1}{2} = 5(x - 1) \)[/tex]
3. [tex]\( y - 1 = \frac{1}{2}(x - 5) \)[/tex]
4. [tex]\( y - 1 = 5\left(x - \frac{1}{2}\right) \)[/tex]
The equation we derived, [tex]\( y - 1 = \frac{1}{2}(x - 5) \)[/tex], matches the third option.
Therefore, the correct equation is:
[tex]\[ y - 1 = \frac{1}{2}(x - 5) \][/tex]
Hence the correct option is the third one, with the numerical result being:
[tex]\[ \boxed{3} \][/tex]
The point-slope form of the equation of a line is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Where:
- [tex]\((x_1, y_1)\)[/tex] is a point on the line.
- [tex]\(m\)[/tex] is the slope of the line.
In this case:
- The point [tex]\((x_1, y_1)\)[/tex] is [tex]\((5, 1)\)[/tex].
- The slope [tex]\(m\)[/tex] is [tex]\(\frac{1}{2}\)[/tex].
Substituting the given values into the point-slope form equation, we get:
[tex]\[ y - 1 = \frac{1}{2}(x - 5) \][/tex]
Now, let's compare this with the given options:
1. [tex]\( y - 5 = \frac{1}{2}(x - 1) \)[/tex]
2. [tex]\( y - \frac{1}{2} = 5(x - 1) \)[/tex]
3. [tex]\( y - 1 = \frac{1}{2}(x - 5) \)[/tex]
4. [tex]\( y - 1 = 5\left(x - \frac{1}{2}\right) \)[/tex]
The equation we derived, [tex]\( y - 1 = \frac{1}{2}(x - 5) \)[/tex], matches the third option.
Therefore, the correct equation is:
[tex]\[ y - 1 = \frac{1}{2}(x - 5) \][/tex]
Hence the correct option is the third one, with the numerical result being:
[tex]\[ \boxed{3} \][/tex]
