To find the equation of the line that is parallel to the line [tex]\( y - 1 = 4(x + 3) \)[/tex] and passes through the point [tex]\( (4, 32) \)[/tex], follow these steps:
1. Identify the slope of the given line. The equation [tex]\( y - 1 = 4(x + 3) \)[/tex] is in point-slope form, [tex]\( y - y_1 = m(x - x_1) \)[/tex], where [tex]\( m \)[/tex] represents the slope.
[tex]\[
y - 1 = 4(x + 3)
\][/tex]
The slope [tex]\( m \)[/tex] here is 4.
2. Since the line we are seeking is parallel to the given line, it will have the same slope, which is [tex]\( m = 4 \)[/tex].
3. Use the point-slope form to find the equation of the new line using the slope [tex]\( m = 4 \)[/tex] and the point [tex]\( (4, 32) \)[/tex]:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Substituting the point [tex]\((4, 32)\)[/tex] and the slope [tex]\( m = 4 \)[/tex]:
[tex]\[
y - 32 = 4(x - 4)
\][/tex]
4. Distribute the slope on the right-hand side of the equation:
[tex]\[
y - 32 = 4x - 16
\][/tex]
5. Solve for [tex]\( y \)[/tex] to convert the equation to slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[
y - 32 = 4x - 16
\][/tex]
Add 32 to both sides:
[tex]\[
y = 4x - 16 + 32
\][/tex]
Simplify:
[tex]\[
y = 4x + 16
\][/tex]
Thus, the equation of the line that is parallel to [tex]\( y - 1 = 4(x + 3) \)[/tex] and passes through the point [tex]\( (4, 32) \)[/tex] is:
[tex]\[
y = 4x + 16
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{y = 4x + 16}
\][/tex]
