To find the equation of a line that is parallel to the given line [tex]\( y = 8x - 1 \)[/tex] and passes through the point [tex]\((-2, 2)\)[/tex], we can follow these steps:
1. Identify the Slope of the Given Line:
The equation of the given line is in the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept. From the equation [tex]\( y = 8x - 1 \)[/tex], we see that the slope [tex]\( m \)[/tex] is 8.
2. Use the Point-Slope Form:
The point-slope form of a line equation is [tex]\( y - y_1 = m(x - x_1) \)[/tex], where [tex]\((x_1, y_1)\)[/tex] is a point on the line and [tex]\( m \)[/tex] is the slope. Here, [tex]\((x_1, y_1) = (-2, 2)\)[/tex] and [tex]\( m = 8 \)[/tex].
Plugging in these values, we get:
[tex]\[
y - 2 = 8(x + 2)
\][/tex]
3. Simplify the Equation:
Expand and simplify the equation:
[tex]\[
y - 2 = 8x + 16
\][/tex]
[tex]\[
y = 8x + 18
\][/tex]
4. Convert to Standard Form:
The standard form of a linear equation is [tex]\( Ax + By = C \)[/tex]. We rearrange the simplified equation [tex]\( y = 8x + 18 \)[/tex] into this form:
[tex]\[
y = 8x + 18
\][/tex]
Subtract [tex]\( 8x \)[/tex] from both sides:
[tex]\[
-8x + y = 18
\][/tex]
5. Identify the Correct Option:
The resulting equation [tex]\(-8x + y = 18\)[/tex] matches the second option:
[tex]\[
-8x + y = 18
\][/tex]
Therefore, the equation of the line parallel to [tex]\( y = 8x - 1 \)[/tex] that passes through the point [tex]\((-2, 2)\)[/tex] is [tex]\(-8x + y = 18\)[/tex]. The correct answer is:
[tex]\[
-8 x + y = 18
\][/tex]
