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.
- Subtract \(4x\) from both sides: \[ 5y = -4x + 7 \]
- 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:
- The equation of the parallel line is \(y = -\frac{4}{5}x + \frac{6}{5}\).
- The equation of the perpendicular line is \(y = \frac{5}{4}x - 7\).