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Consider the line [tex]\( y = -\frac{9}{8} x + 5 \)[/tex].

1. Find the equation of the line that is perpendicular to this line and passes through the point [tex]\( (8, -3) \)[/tex].
2. Find the equation of the line that is parallel to this line and passes through the point [tex]\( (8, -3) \)[/tex].

Note that the ALEKS graphing calculator may be helpful in checking your answer.

Equation of perpendicular line: [tex]\( \boxed{} \)[/tex]

Equation of parallel line: [tex]\( \boxed{} \)[/tex]


Sagot :

To address the problem systematically, let's break it down step-by-step for both perpendicular and parallel lines.

### Equation of the Perpendicular Line

1. Identify the slope of the original line:

The given line is [tex]\( y = -\frac{9}{8}x + 5 \)[/tex].
From this, we can see that the slope [tex]\( m \)[/tex] is [tex]\( -\frac{9}{8} \)[/tex].

2. Determine the slope of the perpendicular line:

The slope of a line perpendicular to another is the negative reciprocal of the slope of the original line. So, for a slope [tex]\( -\frac{9}{8} \)[/tex], the negative reciprocal is:
[tex]\[ \text{slope perpendicular} = -\left(\frac{1}{-\frac{9}{8}}\right) = \frac{8}{9} \][/tex]

3. Use the point-slope form to find equation:

The point-slope form of a line is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Here, the point given is [tex]\( (8, -3) \)[/tex] and the slope [tex]\( m \)[/tex] is [tex]\( \frac{8}{9} \)[/tex]. Plugging these values in:
[tex]\[ y - (-3) = \frac{8}{9}(x - 8) \][/tex]
Simplifying:
[tex]\[ y + 3 = \frac{8}{9}(x - 8) \][/tex]
So the equation of the perpendicular line is:
[tex]\[ \boxed{y + 3 = \frac{8}{9}(x - 8)} \][/tex]

### Equation of the Parallel Line

1. Identify the slope of the parallel line:

The slope of a parallel line will be the same as the original line. Therefore, it remains:
[tex]\[ \text{slope parallel} = -\frac{9}{8} \][/tex]

2. Use the point-slope form to find equation:

Again using the point-slope formula with the given point [tex]\( (8, -3) \)[/tex] and the slope [tex]\( -\frac{9}{8} \)[/tex]:
[tex]\[ y - (-3) = -\frac{9}{8}(x - 8) \][/tex]
Simplifying:
[tex]\[ y + 3 = -\frac{9}{8}(x - 8) \][/tex]
Therefore, the equation of the parallel line is:
[tex]\[ \boxed{y + 3 = -\frac{9}{8}(x - 8)} \][/tex]

In summary:
- The equation of the perpendicular line is:
[tex]\[ \boxed{y + 3 = \frac{8}{9}(x - 8)} \][/tex]
- The equation of the parallel line is:
[tex]\[ \boxed{y + 3 = -\frac{9}{8}(x - 8)} \][/tex]
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