Question

Which line is perpendicular to the line y=−7/4x and goes through the point (−7,5)?(1 point)
Responses

y=−7/4x+5
y equals negative Start Fraction 7 over 4 End Fraction x plus 5

y=4/7x+9
y equals Start Fraction 4 over 7 End Fraction x plus 9

y=−4/7x+1
y equals negative Start Fraction 4 over 7 End Fraction x plus 1

y=4/7x+5

Answer 1

To find the equation of the line that is perpendicular to the 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. **Find 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 to find the equation of the line with the slope \( \frac{4}{7} \) that passes through the point \((-7, 5)\):**
The point-slope form is given by:
\[
y - y_1 = m(x - x_1)
\]
Here, \( (x_1, y_1) = (-7, 5) \) and \( m = \frac{4}{7} \). Plugging in these values:
\[
y - 5 = \frac{4}{7}(x + 7)
\]

4. **Simplify the equation:**
First, expand the right side:
\[
y - 5 = \frac{4}{7}x + \frac{4}{7} \times 7
\]
\[
y - 5 = \frac{4}{7}x + 4
\]
Next, add 5 to both sides:
\[
y = \frac{4}{7}x + 4 + 5
\]
\[
y = \frac{4}{7}x + 9
\]

So, the equation of the line that is perpendicular to \( y = -\frac{7}{4}x \) and passes through the point \((-7, 5)\) is:
\[
y = \frac{4}{7}x + 9
\]

Therefore, the correct response is:
**y = \frac{4}{7}x + 9**.