Question

Equation for a line that passes through (-5, -1) and (5, -3)

Answer 1

To find the equation of the line that passes through the points \((-5, -1)\) and \( (5, -3)\), we can use the two-point form of the equation of a line, which requires us to first calculate the slope (m) of the line.

The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

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

Here, let:

\((x_1, y_1) = (-5, -1)\) and \((x_2, y_2) = (5, -3)\).

Substituting these values into the slope formula:

\[
m = \frac{-3 - (-1)}{5 - (-5)} = \frac{-3 + 1}{5 + 5} = \frac{-2}{10} = -\frac{1}{5}
\]

Now that we know the slope is \(-\frac{1}{5}\), we can use the point-slope form of the equation of a line, which is:

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

Using point \((-5, -1)\) and the slope \(m = -\frac{1}{5}\):

\[
y - (-1) = -\frac{1}{5}(x - (-5))
\]
\[
y + 1 = -\frac{1}{5}(x + 5)
\]

Now, simplify the equation:

\[
y + 1 = -\frac{1}{5}x - 1
\]

Subtracting 1 from both sides gives:

\[
y = -\frac{1}{5}x - 2
\]

Thus, the equation of the line in slope-intercept form (\(y = mx + b\)) is:

\[
y = -\frac{1}{5}x - 2
\]

This is the equation of the line that passes through the points \((-5, -1)\) and \( (5, -3)\).