To complete the point-slope equation of the line through the given points [tex]\((1,3)\)[/tex] and [tex]\((5,1)\)[/tex], we can follow these steps:
1. Identify the coordinates of the two points:
- [tex]\( (x_1, y_1) = (1, 3) \)[/tex]
- [tex]\( (x_2, y_2) = (5, 1) \)[/tex]
2. Calculate the slope [tex]\( m \)[/tex] of the line:
The formula for the slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting the coordinates, we get:
[tex]\[
m = \frac{1 - 3}{5 - 1} = \frac{-2}{4} = -0.5
\][/tex]
3. Use the point-slope form of a line equation:
The point-slope form of a line with slope [tex]\( m \)[/tex] through a point [tex]\((x_1, y_1)\)[/tex] is:
[tex]\[
y - y_1 = m (x - x_1)
\][/tex]
Substituting [tex]\( m = -0.5 \)[/tex] and the point [tex]\((1, 3)\)[/tex], we get:
[tex]\[
y - 3 = -0.5 (x - 1)
\][/tex]
Therefore, the completed point-slope equation of the line through the points [tex]\((1, 3)\)[/tex] and [tex]\((5, 1)\)[/tex] is:
[tex]\[
y - 3 = -0.5 (x - 1)
\][/tex]
This simplified point-slope form gives the exact equation of the line passing through the specified points.
