Question

Which line is perpendicular for the line y equals negative 7/4x and goes through the point negative 7, 5

Answer 1

To find the line that is perpendicular to the given line \( y = -\frac{7}{4}x \) and passes through the point \((-7, 5)\), we need to follow these steps:

1. **Determine the slope of the given line.**
The slope of the line \( y = -\frac{7}{4}x \) is \(-\frac{7}{4}\).

2. **Calculate the slope of the perpendicular line.**
The slope of a line that is perpendicular to another line is the negative reciprocal of the original line's slope. Therefore, the slope \( m \) of the perpendicular line is:
\[
m = -\frac{1}{-\frac{7}{4}} = \frac{4}{7}
\]

3. **Use the point-slope form of a line to write the equation.**
The point-slope form is given by:
\[
y - y_1 = m(x - x_1)
\]
where \( (x_1, y_1) \) is the point \((-7, 5)\) and \( m \) is the slope \( \frac{4}{7} \).

Plugging in the values:
\[
y - 5 = \frac{4}{7}(x + 7)
\]

4. **Distribute and simplify the equation.**
\[
y - 5 = \frac{4}{7}x + \frac{4}{7} \times 7
\]
\[
y - 5 = \frac{4}{7}x + 4
\]
\[
y = \frac{4}{7}x + 4 + 5
\]
\[
y = \frac{4}{7}x + 9
\]

Thus, the equation of the line that is perpendicular to \( y = -\frac{7}{4}x \) and passes through the point \((-7, 5)\) is

\[
y = \frac{4}{7}x + 9.
\]