To determine which equation represents the line that passes through the points [tex]\((0, 6)\)[/tex] and [tex]\((2, 0)\)[/tex], we follow these steps:
1. Find the slope (m):
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 given by:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Plugging in the coordinates [tex]\((x_1, y_1) = (0, 6)\)[/tex] and [tex]\((x_2, y_2) = (2, 0)\)[/tex]:
[tex]\[
m = \frac{0 - 6}{2 - 0} = \frac{-6}{2} = -3
\][/tex]
2. Find the y-intercept (b):
The y-intercept is the value of [tex]\(y\)[/tex] when [tex]\(x = 0\)[/tex]. Since one of our points is [tex]\((0, 6)\)[/tex], the y-intercept is clearly:
[tex]\[
b = 6
\][/tex]
3. Construct the equation of the line:
The general form of the equation of a line is:
[tex]\[
y = mx + b
\][/tex]
Substituting the slope ([tex]\(m = -3\)[/tex]) and the y-intercept ([tex]\(b = 6\)[/tex]) into this formula:
[tex]\[
y = -3x + 6
\][/tex]
4. Match the equation with the given choices:
From the choices provided:
[tex]\[
1. \quad y = -\frac{1}{3}x + 2
\][/tex]
[tex]\[
2. \quad y = -\frac{1}{3}x + 6
\][/tex]
[tex]\[
3. \quad y = -3x + 2
\][/tex]
[tex]\[
4. \quad y = -3x + 6
\][/tex]
It becomes clear that the correct equation is:
[tex]\[
\boxed{y = -3x + 6}
\][/tex]
Thus, the correct equation that represents the line passing through the points [tex]\((0, 6)\)[/tex] and [tex]\((2, 0)\)[/tex] is:
[tex]\(\boxed{y = -3x + 6}\)[/tex]
Therefore, the answer is the 4th choice.
