To find the equation of a line parallel to line [tex]\( AB \)[/tex] which passes through the point [tex]\((4, 5)\)[/tex], follow these steps:
1. Identify the slope of the given line:
The given line [tex]\( y = 5x + 1 \)[/tex] is in slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope.
Thus, the slope [tex]\( m \)[/tex] of line [tex]\( AB \)[/tex] is [tex]\( 5 \)[/tex].
2. Parallel lines have the same slope:
Since the lines are parallel, the slope of the new line will also be [tex]\( 5 \)[/tex].
3. Use the point-slope form to formulate the new line:
The point-slope form of a line is given by:
[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, the point [tex]\((4, 5)\)[/tex] is on the new line, and the slope [tex]\( m \)[/tex] is [tex]\( 5 \)[/tex].
Substituting the values, the point-slope form becomes:
[tex]\[
y - 5 = 5(x - 4)
\][/tex]
4. Simplify to get the slope-intercept form:
Distribute the slope [tex]\( 5 \)[/tex] on the right side:
[tex]\[
y - 5 = 5x - 20
\][/tex]
Add [tex]\( 5 \)[/tex] to both sides to isolate [tex]\( y \)[/tex]:
[tex]\[
y = 5x - 20 + 5
\][/tex]
[tex]\[
y = 5x - 15
\][/tex]
Thus, the equation of the line parallel to line [tex]\( AB \)[/tex] that passes through the point [tex]\((4, 5)\)[/tex] is [tex]\( y = 5x - 15 \)[/tex].
Therefore, the correct choice is:
[tex]\[ \boxed{y = 5x - 15} \][/tex]
