Question

Which line is perpendicular to the line y=−7/4x and goes through the point (−7,5)

Answer 1

To find a line that is perpendicular to the line given by the equation \( y = -\frac{7}{4}x \) and passes through the point \((-7, 5)\), we need to determine the slope of the perpendicular line.

1. **Identify the slope of the original line:**
The slope (m) of the line \( y = -\frac{7}{4}x \) is \( -\frac{7}{4} \).

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

3. **Use the point-slope form of the line equation:**
We use the point-slope form of the line equation, which is given by:
\[
y - y_1 = m(x - x_1)
\]
where \((x_1, y_1)\) is the point through which the line passes (in this case, \((-7, 5)\)), and \(m\) is the slope.

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

4. **Simplify the equation:**
This gives us:
\[
y - 5 = \frac{4}{7}(x + 7)
\]
Now, distribute:
\[
y - 5 = \frac{4}{7}x + 4
\]
Now add 5 to both sides:
\[
y = \frac{4}{7}x + 4 + 5
\]
\[
y = \frac{4}{7}x + 9
\]

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

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