Question

A line passes through the points (2,–5) and (5,–6). What is its equation in point-slope form?

Answer 1

To find the equation of the line in point-slope form, we first need to determine the slope of the line that passes through the points (2, –5) and (5, –6).

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

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

Using the points \((2, -5)\) and \((5, -6)\):

- Let \((x_1, y_1) = (2, -5)\)
- Let \((x_2, y_2) = (5, -6)\)

Now, plug these values into the slope formula:

\[
m = \frac{-6 - (-5)}{5 - 2} = \frac{-6 + 5}{5 - 2} = \frac{-1}{3}
\]

Thus, the slope \( m \) is \(-\frac{1}{3}\).

Next, we can use the point-slope form of the equation of a line, which is:

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

We can choose either of the two points to use in this formula. Let's use the point (2, -5):

- \( (x_1, y_1) = (2, -5) \)
- \( m = -\frac{1}{3} \)

Substituting these values into the point-slope form, we get:

\[
y - (-5) = -\frac{1}{3}(x - 2)
\]

This simplifies to:

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

So, the equation of the line in point-slope form is:

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