To determine which equation is not the equation of the line passing through the points [tex]\((3, 6)\)[/tex] and [tex]\((1, -2)\)[/tex], let's analyze the situation in detail.
1. Calculate the slope ([tex]\(m\)[/tex]) of the line passing through the points [tex]\((3, 6)\)[/tex] and [tex]\((1, -2)\)[/tex].
The formula for the slope [tex]\(m\)[/tex] is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Plugging in the coordinates:
[tex]\[
m = \frac{-2 - 6}{1 - 3} = \frac{-8}{-2} = 4
\][/tex]
2. Find the equation of the line 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]
Using the slope [tex]\(m = 4\)[/tex] and the point [tex]\((3, 6)\)[/tex]:
[tex]\[
y - 6 = 4(x - 3)
\][/tex]
Simplifying this equation to the slope-intercept form [tex]\(y = mx + b\)[/tex]:
[tex]\[
y - 6 = 4(x - 3)
\][/tex]
[tex]\[
y - 6 = 4x - 12
\][/tex]
[tex]\[
y = 4x - 6
\][/tex]
So, the equation of the line is [tex]\(y = 4x - 6\)[/tex].
3. Verify each given choice by simplifying to see if it matches [tex]\(y = 4x - 6\)[/tex]:
A. [tex]\(y - 6 = 4(x - 3)\)[/tex]
[tex]\[
y - 6 = 4x - 12
\][/tex]
[tex]\[
y = 4x - 6
\][/tex]
This matches the line's equation [tex]\(y = 4x - 6\)[/tex].
B. [tex]\(y + 2 = 4(x - 1)\)[/tex]
[tex]\[
y + 2 = 4x - 4
\][/tex]
[tex]\[
y = 4x - 6
\][/tex]
This matches the line's equation [tex]\(y = 4x - 6\)[/tex].
C. [tex]\(y - 2 = 4(x + 1)\)[/tex]
[tex]\[
y - 2 = 4x + 4
\][/tex]
[tex]\[
y = 4x + 6
\][/tex]
This does NOT match the line's equation [tex]\(y = 4x - 6\)[/tex].
D. [tex]\(y = 4x - 6\)[/tex]
This is already in the correct form and matches the line's equation [tex]\(y = 4x - 6\)[/tex].
The incorrect choice is C. Therefore, the equation that is not the equation of the line passing through the points [tex]\((3, 6)\)[/tex] and [tex]\((1, -2)\)[/tex] is:
~[tex]\[ \boxed{C} \][/tex]
