Question

13.
A line passes through (3, –2) and (6, 2). Write an equation for the line in point-slope form.
Rewrite the equation in standard form using integers.

Answer 1

To find the equation of the line that passes through the points (3, -2) and (6, 2), we first need to calculate the slope (m) of the line using the formula:

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

Using the points (x₁, y₁) = (3, -2) and (x₂, y₂) = (6, 2):

\[
m = \frac{2 - (-2)}{6 - 3} = \frac{2 + 2}{3} = \frac{4}{3}
\]

Now that we have the slope, we can write the equation of the line in point-slope form, which is given by:

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

Using point (3, -2):

\[
y - (-2) = \frac{4}{3}(x - 3)
\]
This simplifies to:

\[
y + 2 = \frac{4}{3}(x - 3)
\]

Next, we will convert this equation to standard form (Ax + By = C) by rearranging and clearing any fractions. First, let's expand and simplify:

\[
y + 2 = \frac{4}{3}x - \frac{4}{3} \cdot 3
\]
\[
y + 2 = \frac{4}{3}x - 4
\]

Now, to isolate \(y\), we subtract 2 from both sides:

\[
y = \frac{4}{3}x - 6
\]

Next, we'll move all terms to one side to create the standard form. To eliminate the fraction, we can multiply the entire equation by 3:

\[
3y = 4x - 18
\]

Now, rearranging gives:

\[
4x - 3y = 18
\]

Therefore, the equation of the line in standard form is:

\[
4x - 3y = 18
\]

To summarize, the final equations are:
- Point-slope form: \(y + 2 = \frac{4}{3}(x - 3)\)
- Standard form: \(4x - 3y = 18\)