Question

Given that the points (5,5) and (-5,1) are on a line, write an equation for that line in point-slope form. Choose whatever points you'd like to use. Please describe whatever method you have used in complete sentences.

Answer 1

To write the equation of the line that passes through the points (5,5) and (-5,1) in point-slope form, we first need to determine the slope of the line using these two points.

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

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

Let's assign the points as follows:
- \((x_1, y_1) = (5, 5)\)
- \((x_2, y_2) = (-5, 1)\)

Now, we can plug these values into the slope formula:

\[
m = \frac{1 - 5}{-5 - 5} = \frac{-4}{-10} = \frac{2}{5}
\]

Now that we have the slope \( m = \frac{2}{5} \), we can use the point-slope form of the equation for a line, which is given by:

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

We can use either of the given points to write the equation. Let's use the point (5, 5).

Substituting \( m = \frac{2}{5} \), and \( (x_1, y_1) = (5, 5) \) into the point-slope form:

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

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

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

In conclusion, to find the equation, I calculated the slope between the two points using the slope formula, and then used that slope in the point-slope form equation with one of the points.