Question

perpendicular to y=2x+9 and has the point (4,-1)

Answer 1

To find the equation of a line that is perpendicular to the line given by \( y = 2x + 9 \) and passes through the point \( (4, -1) \), we can follow these steps:

1. **Determine the slope of the given line:**
The slope of the line \( y = 2x + 9 \) is \( 2 \).

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:
\[
m = -\frac{1}{2}
\]

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 (in this case, \( (4, -1) \)) and \( m \) is the slope.

4. **Substituting the values into the point-slope form:**
\[
y - (-1) = -\frac{1}{2}(x - 4)
\]
Simplifying this:
\[
y + 1 = -\frac{1}{2}(x - 4)
\]
\[
y + 1 = -\frac{1}{2}x + 2
\]
\[
y = -\frac{1}{2}x + 1
\]

5. **Write the final equation:**
The equation of the line that is perpendicular to \( y = 2x + 9 \) and passes through the point \( (4, -1) \) is:
\[
y = -\frac{1}{2}x + 1
\]