To identify an equation in slope-intercept form for the line that is parallel to the given line [tex]\( y = 5x + 2 \)[/tex] and passes through the point [tex]\((-6, -1)\)[/tex], follow these steps:
1. Determine the slope of the given line:
The given line is [tex]\( y = 5x + 2 \)[/tex], which is already in slope-intercept form [tex]\( y = mx + b \)[/tex]. Here, [tex]\( m \)[/tex] (the slope) is 5.
2. Use the point-slope form for the new line:
Since parallel lines have the same slope, the slope [tex]\( m \)[/tex] of the desired line is also 5. We need to find the equation of the line that passes through the point [tex]\((-6, -1)\)[/tex].
The point-slope form of a line is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Here, [tex]\((x_1, y_1)\)[/tex] is the point [tex]\((-6, -1)\)[/tex] and [tex]\( m = 5 \)[/tex].
3. Substitute the given point and slope into the point-slope form:
[tex]\[
y - (-1) = 5(x - (-6))
\][/tex]
Simplify:
[tex]\[
y + 1 = 5(x + 6)
\][/tex]
4. Distribute and simplify to get the slope-intercept form:
[tex]\[
y + 1 = 5x + 30
\][/tex]
[tex]\[
y = 5x + 30 - 1
\][/tex]
[tex]\[
y = 5x + 29
\][/tex]
So, the equation of the line parallel to [tex]\( y = 5x + 2 \)[/tex] and passing through [tex]\((-6, -1)\)[/tex] is [tex]\( y = 5x + 29 \)[/tex].
The correct choice from the given options is:
D. [tex]\( y = 5x + 29 \)[/tex]
