Question

What point would a line with the following description pass through?
The line passes through (-1, -3) and is perpendicular to a line that passes through (-5, -4) and (5, -6)

Answer 1

To find the point through which the line passes, we need to determine the slope of the given line that passes through the points (-5, -4) and (5, -6).

1. **Calculate the slope of the line through (-5, -4) and (5, -6)**:
The formula for the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Plugging in the coordinates of the points:
\[
m = \frac{-6 - (-4)}{5 - (-5)} = \frac{-6 + 4}{5 + 5} = \frac{-2}{10} = -\frac{1}{5}
\]

2. **Find the slope of the line perpendicular to it**:
The slope of a line that is perpendicular to another line is the negative reciprocal of the original slope. The slope we just found is \( -\frac{1}{5} \), so the slope of the perpendicular line \( m' \) is:
\[
m' = -\frac{1}{-\frac{1}{5}} = 5
\]

3. **Use the point-slope form of the equation of a line**:
The point-slope form of a line 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 will use the point (-1, -3) and the slope we calculated (5):
\[
y - (-3) = 5(x - (-1))
\]
This simplifies to:
\[
y + 3 = 5(x + 1)
\]
Distributing the 5 gives:
\[
y + 3 = 5x + 5
\]
Subtracting 3 from both sides results in:
\[
y = 5x + 2
\]

4. **Identify a point through which the line passes**:
Since the original point through which we want to find the line is already given as (-1, -3), we have confirmed that the line passes through this point.

Thus, the answer is the point **(-1, -3)**.