To determine the equation of a line parallel to the line [tex]\(y = 8x - 1\)[/tex] that passes through the point [tex]\((0, 2)\)[/tex], we need to follow these steps:
1. Identify the slope of the given line:
The given line is [tex]\(y = 8x - 1\)[/tex]. The slope-intercept form of a line is [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope.
Here, the slope [tex]\(m\)[/tex] is 8.
2. Use the point-slope form of the equation of a line:
The point-slope form is [tex]\(y - y_1 = m(x - x_1)\)[/tex], where [tex]\((x_1, y_1)\)[/tex] is the point through which the line passes.
We are given the point [tex]\((0, 2)\)[/tex].
3. Substitute the point [tex]\((0, 2)\)[/tex] and the slope [tex]\(8\)[/tex] into the point-slope form:
[tex]\[
y - 2 = 8(x - 0)
\][/tex]
4. Simplify the equation:
[tex]\[
y - 2 = 8x
\][/tex]
[tex]\[
y = 8x + 2
\][/tex]
5. Convert this equation to standard form [tex]\(Ax + By = C\)[/tex]:
We need to rearrange [tex]\(y = 8x + 2\)[/tex] into the form [tex]\(Ax + By = C\)[/tex].
Subtract [tex]\(8x\)[/tex] from both sides to get:
[tex]\[
-8x + y = 2
\][/tex]
6. Adjust the coefficients to match the standard form criteria (A should be positive):
Multiply the entire equation by [tex]\(-1\)[/tex]:
[tex]\[
8x - y = -2
\][/tex]
Therefore, the equation of the line in standard form, parallel to [tex]\(y = 8x - 1\)[/tex] and passing through the point [tex]\((0, 2)\)[/tex], is:
[tex]\[
8x - y = -2
\][/tex]
So, the correct answer is:
[tex]\[
8x - y = -2
\][/tex]
