Question

Find the equation of a line perpendicular to y+2=1/2x that passes through the point (-4,-2)

Answer 1

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

1. **Find the slope of the given line**:
The equation is in point-slope form, where the slope (\(m\)) can be extracted. The equation \( y + 2 = \frac{1}{2}x \) implies that the slope \(m = \frac{1}{2}\).

2. **Determine the slope of the perpendicular line**:
The slope of a line perpendicular to another is the negative reciprocal of the original slope. Therefore:
\[
m_{\text{perpendicular}} = -\frac{1}{m} = -\frac{1}{\frac{1}{2}} = -2
\]

3. **Use the point-slope form to find the equation of the perpendicular line**:
The point-slope form of the line can be given by the formula:
\[
y - y_1 = m (x - x_1)
\]
where \( (x_1, y_1) \) is the point through which the line passes and \( m \) is the slope. Substituting the point \((-4, -2)\) and the slope \(-2\):
\[
y - (-2) = -2(x - (-4))
\]

Simplifying this:
\[
y + 2 = -2(x + 4)
\]
\[
y + 2 = -2x - 8
\]
\[
y = -2x - 8 - 2
\]
\[
y = -2x - 10
\]

Thus, the equation of the line that is perpendicular to \(y + 2 = \frac{1}{2}x\) and passes through the point \((-4, -2)\) is:
\[
\boxed{y = -2x - 10}
\]