To find the equation of the line passing through the points [tex]\((0,6)\)[/tex] and [tex]\((2,10)\)[/tex], we follow these steps:
1. Determine the slope ([tex]\(m\)[/tex]) of the line:
The slope of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by the formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting the given points [tex]\((x_1, y_1) = (0, 6)\)[/tex] and [tex]\((x_2, y_2) = (2, 10)\)[/tex], we get:
[tex]\[
m = \frac{10 - 6}{2 - 0} = \frac{4}{2} = 2
\][/tex]
2. Find the y-intercept ([tex]\(b\)[/tex]):
The slope-intercept form of a line's equation is:
[tex]\[
y = mx + b
\][/tex]
To find [tex]\(b\)[/tex], we can substitute one of the points and the slope into the slope-intercept form equation. Let's use the point [tex]\((0,6)\)[/tex]:
[tex]\[
6 = 2(0) + b
\][/tex]
This simplifies to:
[tex]\[
6 = b
\][/tex]
So, the y-intercept [tex]\(b = 6\)[/tex].
3. Write the equation of the line:
Now that we have the slope [tex]\(m = 2\)[/tex] and the y-intercept [tex]\(b = 6\)[/tex], we can write the equation of the line as:
[tex]\[
y = 2x + 6
\][/tex]
Therefore, the equation of the line that passes through the points [tex]\((0, 6)\)[/tex] and [tex]\((2, 10)\)[/tex] is:
[tex]\[
\boxed{y = 2x + 6}
\][/tex]
Consequently, the correct choice is:
A. [tex]\(y = 2x + 6\)[/tex]
