Question

Select the equation that is perpendicular to

y=(3/4)x+6 and goes through the point (3,8)

a
y=(3/4)x-2

b
y=(-4/3)x-10

c
y=(-4/3)x+12

Answer 1

To determine the equation that is perpendicular to \( y = \frac{3}{4}x + 6 \) and passes through the point \( (3, 8) \), we first need to find the slope of the perpendicular line.

1. **Identify the slope of the original line**: The slope of \( y = \frac{3}{4}x + 6 \) is \( \frac{3}{4} \).

2. **Find the slope of the perpendicular line**: The slope of a line perpendicular to another line is the negative reciprocal of the original slope. Therefore, the slope of the line we need is:
\[
m = -\frac{1}{m_{original}} = -\frac{1}{\frac{3}{4}} = -\frac{4}{3}
\]

3. **Use the point-slope form of a line**: We can use the point-slope formula \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) = (3, 8) \) and \( m = -\frac{4}{3} \):
\[
y - 8 = -\frac{4}{3}(x - 3)
\]

4. **Simplify the equation**:
\[
y - 8 = -\frac{4}{3}x + 4
\]
\[
y = -\frac{4}{3}x + 12
\]

Now we can compare this equation to the provided options:

a) \( y = \frac{3}{4}x - 2 \)

b) \( y = -\frac{4}{3}x - 10 \)

c) \( y = -\frac{4}{3}x + 12 \)

The correct equation for the line that is perpendicular to the original line and passes through the point \( (3, 8) \) is:

**c) \( y = -\frac{4}{3}x + 12 \)**.