Questions: Consider the line (y=-frac83 x-2) the equation of the line that is parallel to this line and passes through the (p) the equation of the line that is perpendicular to this line and passes through Equation of parallel line: Equation of perpendicular line:

Consider the line (y=-frac83 x-2) the equation of the line that is parallel to this line and passes through the (p) the equation of the line that is perpendicular to this line and passes through Equation of parallel line: Equation of perpendicular line:
Transcript text: der the line $y=-\frac{8}{3} x-2$ the equation of the line that is parallel to this line and passes through the $p$ the equation of the line that is perpendicular to this line and passes through Equation of parallel line: Equation of perpendicular line:
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Solution

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Solution Steps

To find the equation of a line that is parallel to a given line, we need to use the same slope as the given line. For a line that is perpendicular, we use the negative reciprocal of the slope of the given line. We then use the point-slope form of the equation of a line to find the specific equations.

Step 1: Identify the Slope of the Given Line

The given line is \( y = -\frac{8}{3}x - 2 \). The slope of this line is \( -\frac{8}{3} \).

Step 2: Determine the Slope of the Parallel Line

A line parallel to the given line will have the same slope. Therefore, the slope of the parallel line is also \( -\frac{8}{3} \).

Step 3: Determine the Slope of the Perpendicular Line

A line perpendicular to the given line will have a slope that is the negative reciprocal of the given line's slope. The negative reciprocal of \( -\frac{8}{3} \) is \( \frac{3}{8} \).

Step 4: Use the Point-Slope Form to Find the Equations

Assume the point through which the lines pass is \( (1, 2) \).

Parallel Line

Using the point-slope form \( y - y_1 = m(x - x_1) \): \[ y - 2 = -\frac{8}{3}(x - 1) \] Simplifying to slope-intercept form \( y = mx + b \): \[ y = -\frac{8}{3}x + \frac{8}{3} + 2 \] \[ y = -\frac{8}{3}x + \frac{14}{3} \]

Perpendicular Line

Using the point-slope form \( y - y_1 = m(x - x_1) \): \[ y - 2 = \frac{3}{8}(x - 1) \] Simplifying to slope-intercept form \( y = mx + b \): \[ y = \frac{3}{8}x - \frac{3}{8} + 2 \] \[ y = \frac{3}{8}x + \frac{13}{8} \]

Final Answer

\[ \boxed{y = -\frac{8}{3}x + \frac{14}{3}} \] \[ \boxed{y = \frac{3}{8}x + \frac{13}{8}} \]

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