Questions: Write an equation for a line passing through the point (4,-3) that is parallel to the line 4x+5y=8. Then write a second equation for a line passing through the point (4,-3) that is perpendicular to the line 4x+5y=8. The equation of the parallel line is (Simplify your answer. Type your answer in slope-intercept form. Use integers or fractions for any numbers in the expression.) The equation of the perpendicular line is (Simplify your answer. Type your answer in slope-intercept form. Use integers or fractions for any numbers in the expression.)

Write an equation for a line passing through the point (4,-3) that is parallel to the line 4x+5y=8. Then write a second equation for a line passing through the point (4,-3) that is perpendicular to the line 4x+5y=8. The equation of the parallel line is (Simplify your answer. Type your answer in slope-intercept form. Use integers or fractions for any numbers in the expression.) The equation of the perpendicular line is (Simplify your answer. Type your answer in slope-intercept form. Use integers or fractions for any numbers in the expression.)
Transcript text: Write an equation for a line passing through the point $(4,-3)$ that is parallel to the line $4 x+5 y=8$. Then write a second equation for a line passing through the point $(4,-3)$ that is perpendicular to the line $4 x+5 y=8$. The equation of the parallel line is $\square$ (Simplify your answer. Type your answer in slope-intercept form. Use integers or fractions for any numbers in the expression.) The equation of the perpendicular line is $\square$ (Simplify your answer. Type your answer in slope-intercept form. Use integers or fractions for any numbers in the expression.)
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Solution

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Solution Steps

To solve this problem, we need to find the equations of two lines: one parallel and one perpendicular to the given line \(4x + 5y = 8\), both passing through the point \((4, -3)\).

  1. Parallel Line:

    • A line parallel to \(4x + 5y = 8\) will have the same slope. First, convert the given line to slope-intercept form \(y = mx + b\) to find the slope \(m\).
    • Use the point-slope form \(y - y_1 = m(x - x_1)\) with the point \((4, -3)\) to find the equation of the parallel line.

  2. Perpendicular Line:

    • A line perpendicular to \(4x + 5y = 8\) will have a slope that is the negative reciprocal of the slope of the given line.
    • Again, use the point-slope form with the point \((4, -3)\) to find the equation of the perpendicular line.
Step 1: Find the Slope of the Given Line

The equation of the given line is

\[ 4x + 5y = 8 \]

To find the slope, we convert this equation to slope-intercept form \(y = mx + b\):

\[ 5y = -4x + 8 \implies y = -\frac{4}{5}x + \frac{8}{5} \]

Thus, the slope \(m\) of the given line is

\[ m = -\frac{4}{5} \]

Step 2: Equation of the Parallel Line

A line parallel to the given line will have the same slope. Using the point \((4, -3)\) and the slope \(-\frac{4}{5}\), we apply the point-slope form:

\[ y - y_1 = m(x - x_1) \]

Substituting the values:

\[ y - (-3) = -\frac{4}{5}(x - 4) \]

Simplifying this gives:

\[ y + 3 = -\frac{4}{5}x + \frac{16}{5} \]

Rearranging to slope-intercept form:

\[ y = -\frac{4}{5}x + \frac{16}{5} - 3 = -\frac{4}{5}x + \frac{16}{5} - \frac{15}{5} = -\frac{4}{5}x + \frac{1}{5} \]

Step 3: Equation of the Perpendicular Line

The slope of a line perpendicular to the given line is the negative reciprocal of \(-\frac{4}{5}\), which is

\[ \frac{5}{4} \]

Using the point \((4, -3)\) and the slope \(\frac{5}{4}\), we again apply the point-slope form:

\[ y - (-3) = \frac{5}{4}(x - 4) \]

Simplifying this gives:

\[ y + 3 = \frac{5}{4}x - 5 \]

Rearranging to slope-intercept form:

\[ y = \frac{5}{4}x - 5 - 3 = \frac{5}{4}x - 8 \]

Final Answer

The equations of the lines are:

  • The equation of the parallel line is

\[ \boxed{y = -\frac{4}{5}x + \frac{1}{5}} \]

  • The equation of the perpendicular line is

\[ \boxed{y = \frac{5}{4}x - 8} \]

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