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

3 answers

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:

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

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

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:

\[ y = 2x + 5 \]