Question

Find the equation of a line perpendicular toy, equals, x, minus, 3y=x−3that passes through the point left parenthesis, minus, 6, comma, 0, right parenthesis(−6,0).

Answer 1

To find the equation of the line that is perpendicular to the line given by the equation \( y = x - 3 \) and passes through the point \( (-6, 0) \), we will follow these steps:

1. **Determine the slope of the given line**: The equation \( y = x - 3 \) can be rewritten in slope-intercept form \( y = mx + b \), where \( m \) is the slope. The slope \( m \) of this line is \( 1 \).

2. **Find the slope of the perpendicular line**: The slope of a line that is perpendicular to another is the negative reciprocal of the original line's slope. Therefore, if the slope of the original line is \( 1 \), the slope \( m_{\perp} \) of the perpendicular line will be:
\[
m_{\perp} = -\frac{1}{1} = -1.
\]

3. **Use the point-slope form of the line**: We can now use the point-slope form of the equation of a line, which is given by:
\[
y - y_1 = m(x - x_1),
\]
where \( (x_1, y_1) \) is a point on the line and \( m \) is the slope. We have the point \( (-6, 0) \) and the slope \( m_{\perp} = -1 \):
\[
y - 0 = -1(x - (-6)).
\]
Simplifying this:
\[
y = -1(x + 6).
\]
\[
y = -x - 6.
\]

4. **Final equation**: The equation of the line perpendicular to \( y = x - 3 \) that passes through \( (-6, 0) \) is:
\[
y = -x - 6.
\]

This can also be written in standard form \( x + y + 6 = 0 \) if desired.