The equation of line [tex][tex]$AB$[/tex][/tex] is [tex][tex]$y = 5x + 1$[/tex][/tex]. Write an equation of a line parallel to line [tex][tex]$AB$[/tex][/tex] in slope-intercept form that contains point [tex][tex]$(4, 5)$[/tex][/tex].

A. [tex][tex]$y = 5x - 15$[/tex][/tex]
B. [tex][tex]$y = 5x + 15$[/tex][/tex]
C. [tex][tex]$y = \frac{1}{5}x + \frac{21}{5}$[/tex][/tex]
D. [tex][tex]$y = \frac{1}{5}x - \frac{29}{5}$[/tex][/tex]



Answer :

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]