Question

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

A: y = -7/4x +5

B: y = 4/7x + 9

C: y = 4/7x + 5

D: y = -4/7x + 1

Answer 1

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

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 is the negative reciprocal of the slope of the original line. Therefore, the negative reciprocal of \(-\frac{7}{4}\) is:
\[
m = \frac{4}{7}
\]

3. **Use the point-slope form of the equation of a line**:
The point-slope form is given by:
\[
y - y_1 = m(x - x_1)
\]
where \( (x_1, y_1) \) is the point through which the line passes, and \( m \) is the slope.

Substituting \( m = \frac{4}{7} \) and the point \((-7, 5)\):
\[
y - 5 = \frac{4}{7}(x + 7)
\]

4. **Simplify this equation**:
Expanding it, we get:
\[
y - 5 = \frac{4}{7}x + 4
\]
\[
y = \frac{4}{7}x + 9
\]

Now, we can match this to the given options:

- A: \( y = -\frac{7}{4}x + 5 \) (not correct)
- B: \( y = \frac{4}{7}x + 9 \) (this matches)
- C: \( y = \frac{4}{7}x + 5 \) (not correct)
- D: \( y = -\frac{4}{7}x + 1 \) (not correct)

Thus, the correct answer is:

**B: \( y = \frac{4}{7}x + 9 \)**.