Sure, let’s work through this problem step-by-step.
The given equation of the line [tex]\( AB \)[/tex] is:
[tex]\[ y = 5x + 1 \][/tex]
Lines that are parallel to each other have the same slope. The equation of a line in slope-intercept form is:
[tex]\[ y = mx + b \][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
From the equation [tex]\( y = 5x + 1 \)[/tex], we can see that the slope [tex]\( m \)[/tex] is:
[tex]\[ m = 5 \][/tex]
Since the new line is parallel to the given line, it will have the same slope. Therefore, the slope of the new line will also be:
[tex]\[ m = 5 \][/tex]
Now, we need to find the y-intercept[tex]\( b \)[/tex] of the new line. We know that the line passes through the point [tex]\( (4, 5) \)[/tex]. We can substitute [tex]\( x = 4 \)[/tex] and [tex]\( y = 5 \)[/tex] into the slope-intercept form [tex]\( y = 5x + b \)[/tex] to find [tex]\( b \)[/tex].
Substitute the values:
[tex]\[ 5 = 5(4) + b \][/tex]
Simplify the equation:
[tex]\[ 5 = 20 + b \][/tex]
Solve for [tex]\( b \)[/tex]:
[tex]\[ b = 5 - 20 \][/tex]
[tex]\[ b = -15 \][/tex]
Thus, the y-intercept [tex]\( b \)[/tex] is:
[tex]\[ b = -15 \][/tex]
Therefore, the equation of the line parallel to [tex]\( AB \)[/tex] and passing through the point [tex]\( (4, 5) \)[/tex] is:
[tex]\[ y = 5x - 15 \][/tex]
So, the correct choice is:
[tex]\[ y = 5x - 15 \][/tex]
