Certainly! Let's break this down step-by-step.
### Finding the Equation of the Parallel Line
1. Determine the Slope of the Original Line:
The original line is given by the equation [tex]\( y = 9x - 9 \)[/tex].
The slope (m) of this line is [tex]\( 9 \)[/tex].
2. Slope of the Parallel Line:
Parallel lines have the same slope. Thus, the slope of the parallel line is also [tex]\( 9 \)[/tex].
3. Using the Point to Find the Y-Intercept (b):
We know the line passes through the point [tex]\( (8, -3) \)[/tex].
The equation of the line in slope-intercept form is [tex]\( y = mx + b \)[/tex].
Substituting [tex]\( m = 9 \)[/tex] and the point [tex]\( (8, -3) \)[/tex]:
[tex]\[
-3 = 9(8) + b
\][/tex]
[tex]\[
-3 = 72 + b
\][/tex]
Solving for [tex]\( b \)[/tex]:
[tex]\[
b = -3 - 72
\][/tex]
[tex]\[
b = -75
\][/tex]
Thus, the equation of the parallel line is:
[tex]\[ y = 9x - 75 \][/tex]
### Finding the Equation of the Perpendicular Line
1. Determine the Negative Reciprocal of the Slope:
The slope of the original line is [tex]\( 9 \)[/tex].
The slope of a line perpendicular to this line is the negative reciprocal of [tex]\( 9 \)[/tex], which is [tex]\( -\frac{1}{9} \)[/tex].
2. Using the Point to Find the Y-Intercept (b):
We know the perpendicular line passes through the point [tex]\( (8, -3) \)[/tex].
The equation of the line in slope-intercept form is [tex]\( y = mx + b \)[/tex], where [tex]\( m = -\frac{1}{9} \)[/tex].
Substituting the slope and the point [tex]\( (8, -3) \)[/tex]:
[tex]\[
-3 = -\frac{1}{9}(8) + b
\][/tex]
[tex]\[
-3 = -\frac{8}{9} + b
\][/tex]
Solving for [tex]\( b \)[/tex]:
[tex]\[
b = -3 + \frac{8}{9}
\][/tex]
Converting [tex]\( -3 \)[/tex] to a fraction with the same denominator:
[tex]\[
-3 = -\frac{27}{9}
\][/tex]
Thus:
[tex]\[
b = -\frac{27}{9} + \frac{8}{9}
\][/tex]
[tex]\[
b = -\frac{27 - 8}{9}
\][/tex]
[tex]\[
b = -\frac{19}{9}
\][/tex]
Thus, the equation of the perpendicular line is:
[tex]\[ y = -\frac{1}{9}x - \frac{19}{9} \][/tex]
In decimal form, the equations of the lines are:
1. Equation of the parallel line: [tex]\( y = 9x - 75 \)[/tex]
2. Equation of the perpendicular line: [tex]\( y = -0.1111x - 2.1111 \)[/tex]
So, the final answers are:
- Equation of the parallel line: [tex]\( y = 9x - 75 \)[/tex]
- Equation of the perpendicular line: [tex]\( y = -0.1111x - 2.1111 \)[/tex]
