Question

Given that the points (-1,3) and (2,9) are on a line, write the equation for that line in point-slope form.

Answer 1

Using your answer from the previous problem, write the equation of that same line in slope-intercept form.

Answer 2

To write the equation of the line in point-slope form, we first need to find the slope (m) of the line that passes through the points (-1, 3) and (2, 9). The formula for the slope 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 (-1, 3) as \((x_1, y_1)\) and (2, 9) as \((x_2, y_2)\):

\[
m = \frac{9 - 3}{2 - (-1)} = \frac{6}{3} = 2
\]

Now that we have the slope \(m = 2\), we can use the point-slope form of the line equation, which is:

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

We can choose either point (-1, 3) or (2, 9) to use in our equation. Let's choose the point (-1, 3):

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

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

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

Answer 3

To convert the equation from point-slope form \(y - 3 = 2(x + 1)\) to slope-intercept form \(y = mx + b\), we need to simplify the equation.

Starting with:

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

First, distribute the 2 on the right-hand side:

\[
y - 3 = 2x + 2
\]

Next, add 3 to both sides to isolate \(y\):

\[
y = 2x + 2 + 3
\]

Simplifying the right side gives:

\[
y = 2x + 5
\]

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

\[
y = 2x + 5
\]