To determine the equation of the line that passes through the points [tex]\((1, 6)\)[/tex] and [tex]\((2, 1)\)[/tex], follow these steps:
1. Calculate the slope (m) of the line:
The slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting the given points [tex]\((1, 6)\)[/tex] and [tex]\((2, 1)\)[/tex]:
[tex]\[
m = \frac{1 - 6}{2 - 1} = \frac{-5}{1} = -5
\][/tex]
2. Calculate the y-intercept (b) of the line:
Using the slope-intercept form of a line equation [tex]\( y = mx + b \)[/tex], we can rearrange to solve for [tex]\( b \)[/tex]:
[tex]\[
b = y - mx
\][/tex]
Using one of the given points, say [tex]\((1, 6)\)[/tex]:
[tex]\[
b = 6 - (-5)(1)
\][/tex]
[tex]\[
b = 6 + 5 = 11
\][/tex]
3. Form the equation of the line:
Substitute the calculated slope [tex]\( m \)[/tex] and y-intercept [tex]\( b \)[/tex] into the slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[
y = -5x + 11
\][/tex]
4. Check which option matches this equation:
Given the options:
A. [tex]\( y = -5x + 1 \)[/tex]
B. [tex]\( y = 5x - 1 \)[/tex]
C. [tex]\( y = 2 \)[/tex]
D. [tex]\( y = -5x + 11 \)[/tex]
The equation we found is [tex]\( y = -5x + 11 \)[/tex], which corresponds to Option D.
Therefore, the equation of the line that passes through the points [tex]\((1, 6)\)[/tex] and [tex]\((2, 1)\)[/tex] is:
[tex]\[
\boxed{y = -5x + 11}
\][/tex]
