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Part 1
Write a​ slope-intercept equation for a line passing through the point left parenthesis 4 comma negative 2 right parenthesis(4,−2) that is parallel to the line 4 x plus 5 y equals 74x+5y=7. Then write a second equation for a line passing through the point left parenthesis 4 comma negative 2 right parenthesis(4,−2) that is perpendicular to the line 4 x plus 5 y equals 74x+5y=7.
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Part 1
The equation of the parallel line is enter your response here.

1 answer

To find the equations of lines that are parallel and perpendicular to a given line, we first need to determine the slope of the given line.

Step 1: Find the Slope of the Given Line

The original equation is:

\[ 4x + 5y = 7 \]

To find the slope, we can rearrange this into slope-intercept form \(y = mx + b\), where \(m\) is the slope.

  1. Subtract \(4x\) from both sides: \[ 5y = -4x + 7 \]
  2. Divide everything by \(5\): \[ y = -\frac{4}{5}x + \frac{7}{5} \]

From this, we see that the slope \(m\) of the line is \(-\frac{4}{5}\).

Step 2: Equation of the Parallel Line

Since parallel lines have the same slope, the slope of the line we want to find will also be \(-\frac{4}{5}\). We will use the point \((4, -2)\) and the point-slope form of the equation of a line:

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

Using \((x_1, y_1) = (4, -2)\) and \(m = -\frac{4}{5}\):

\[ y - (-2) = -\frac{4}{5}(x - 4) \]

Now, simplify this equation:

\[ y + 2 = -\frac{4}{5}x + \frac{16}{5} \]

Subtract 2 from both sides (which is the same as \(-\frac{10}{5}\)):

\[ y = -\frac{4}{5}x + \frac{16}{5} - \frac{10}{5} \] \[ y = -\frac{4}{5}x + \frac{6}{5} \]

Thus, the equation of the parallel line is:

\[ \boxed{y = -\frac{4}{5}x + \frac{6}{5}} \]

Step 3: Equation of the Perpendicular Line

To find the slope of the line that is perpendicular to the original line, we take the negative reciprocal of the slope \(-\frac{4}{5}\). The negative reciprocal is:

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

Now we will use the point \( (4, -2) \) again and this new slope:

Using \(m = \frac{5}{4}\):

\[ y - (-2) = \frac{5}{4}(x - 4) \]

Simplifying:

\[ y + 2 = \frac{5}{4}x - 5 \]

Subtract \(2\) (which is the same as \(\frac{8}{4}\)) from both sides:

\[ y = \frac{5}{4}x - 5 - 2 \] \[ y = \frac{5}{4}x - 7 \]

Thus, the equation of the perpendicular line is:

\[ \boxed{y = \frac{5}{4}x - 7} \]

In summary:

  1. The equation of the parallel line is \(y = -\frac{4}{5}x + \frac{6}{5}\).
  2. The equation of the perpendicular line is \(y = \frac{5}{4}x - 7\).