Question

What points 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 determine the points that the line passes through, we first need to find the slope of the line that passes through the points (-5, -4) and (5, -6).

The slope \( m \) of a line defined by two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Plugging in our points:

\[
m = \frac{-6 - (-4)}{5 - (-5)} = \frac{-6 + 4}{5 + 5} = \frac{-2}{10} = -\frac{1}{5}
\]

The slope of the line passing through (-5, -4) and (5, -6) is \(-\frac{1}{5}\).

Since the line we want to find is perpendicular to this line, its slope will be the negative reciprocal of \(-\frac{1}{5}\):

\[
m_{\text{perpendicular}} = -\frac{1}{(-\frac{1}{5})} = 5
\]

Next, we have the slope of the line we want to find, which passes through the point (-1, -3) and has a slope of 5. We can use the point-slope form of the equation of a line, which is given by:

\[
y - y_1 = m(x - x_1)
\]

Substituting \((-1, -3)\) for \((x_1, y_1)\) and \(5\) for \(m\):

\[
y - (-3) = 5(x - (-1))
\]
\[
y + 3 = 5(x + 1)
\]
\[
y + 3 = 5x + 5
\]
\[
y = 5x + 5 - 3
\]
\[
y = 5x + 2
\]

Now, with the equation of the line \( y = 5x + 2 \), we can determine points that this line passes through by substituting different x values.

For example:
1. If \( x = 0 \):
\[
y = 5(0) + 2 = 2 \quad \Rightarrow \quad (0, 2)
\]
2. If \( x = 1 \):
\[
y = 5(1) + 2 = 7 \quad \Rightarrow \quad (1, 7)
\]
3. If \( x = -2 \):
\[
y = 5(-2) + 2 = -10 + 2 = -8 \quad \Rightarrow \quad (-2, -8)
\]

Therefore, the line \( y = 5x + 2 \) passes through points such as \((-1, -3)\), \((0, 2)\), \((1, 7)\), and \((-2, -8)\), among many others. All points of the form \((x, 5x + 2)\) where \(x\) is any real number lie on this line.