To solve this problem, let's follow the step-by-step approach to find the equation of the line that is perpendicular to [tex]\( y = 4x + 5 \)[/tex] and passes through the point [tex]\((8, -4)\)[/tex].
1. Identify the slope of the original line:
The equation of the given line is [tex]\( y = 4x + 5 \)[/tex]. In slope-intercept form [tex]\( y = mx + b \)[/tex], the coefficient of [tex]\( x \)[/tex] (which is 4) represents the slope ([tex]\( m \)[/tex]). So, the slope of the original line is [tex]\( 4 \)[/tex].
2. Determine the slope of the perpendicular line:
The slope of a line perpendicular to another line is the negative reciprocal of the slope of the original line. If the slope of the original line is [tex]\( m \)[/tex], the slope of the perpendicular line is [tex]\( -\frac{1}{m} \)[/tex].
For the given line with slope [tex]\( 4 \)[/tex], the slope of the perpendicular line will be:
[tex]\[
-\frac{1}{4}
\][/tex]
3. Use the point-slope form to write the equation of the perpendicular line:
The point-slope form of a line's equation 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 (in this case, [tex]\( (8, -4) \)[/tex]), and [tex]\( m \)[/tex] is the slope of the line.
Plugging in the point [tex]\((8, -4)\)[/tex] and the slope [tex]\( -\frac{1}{4} \)[/tex] into the point-slope form:
[tex]\[
y - (-4) = -\frac{1}{4}(x - 8)
\][/tex]
4. Simplify the equation:
Simplify the equation to get it into slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[
y + 4 = -\frac{1}{4}(x - 8)
\][/tex]
Distribute [tex]\(-\frac{1}{4}\)[/tex] on the right-hand side:
[tex]\[
y + 4 = -\frac{1}{4}x + 2
\][/tex]
Isolate [tex]\( y \)[/tex] on the left side:
[tex]\[
y = -\frac{1}{4}x + 2 - 4
\][/tex]
Simplify the constants on the right side:
[tex]\[
y = -\frac{1}{4}x - 2
\][/tex]
The equation of the line that is perpendicular to [tex]\( y = 4x + 5 \)[/tex] and passes through the point [tex]\( (8, -4) \)[/tex] is:
[tex]\[
y = -\frac{1}{4}x - 2
\][/tex]
Therefore:
[tex]\[
y = -\frac{1}{4} x - 2
\][/tex]
The symbol that belongs in the green box is minus.
