To determine the equation of the line that is perpendicular to [tex]\( y = 5x + 4 \)[/tex] and passes through the point [tex]\( (15, -2) \)[/tex], we need to follow these steps:
1. Determine the slope of the original line:
The original line is given by [tex]\( y = 5x + 4 \)[/tex]. The slope [tex]\( m \)[/tex] of this line is 5.
2. Find the slope of the perpendicular line:
The slopes of perpendicular lines are negative reciprocals of each other. Therefore, the slope of the line perpendicular to [tex]\( y = 5x + 4 \)[/tex] is:
[tex]\[
-\frac{1}{5}
\][/tex]
3. Form the equation of the new line:
We now use the point-slope form of a linear equation, which is:
[tex]\[
y - y_1 = m (x - x_1)
\][/tex]
Here, [tex]\( (x_1, y_1) = (15, -2) \)[/tex] and [tex]\( m = -\frac{1}{5} \)[/tex]. Plugging these values into the equation, we get:
[tex]\[
y - (-2) = -\frac{1}{5} (x - 15)
\][/tex]
Simplifying this, we have:
[tex]\[
y + 2 = -\frac{1}{5}x + \frac{15}{5}
\][/tex]
[tex]\[
y + 2 = -\frac{1}{5}x + 3
\][/tex]
4. Solve for [tex]\( b \)[/tex] in the new equation [tex]\( y = -\frac{1}{5}x + b \)[/tex]:
To get the equation in the form [tex]\( y = -\frac{1}{5}x + b \)[/tex], we isolate [tex]\( y \)[/tex]:
[tex]\[
y = -\frac{1}{5}x + 3 - 2
\][/tex]
Simplifying this, we get:
[tex]\[
y = -\frac{1}{5}x + 1
\][/tex]
5. Identify the value of [tex]\( b \)[/tex]:
Comparing this with the form [tex]\( y = -\frac{1}{5}x + b \)[/tex], it is clear that:
[tex]\[
b = 1
\][/tex]
Thus, the value of [tex]\( b \)[/tex] is [tex]\( \boxed{1} \)[/tex].
