To start, we’ll find the slope of the line passing through the points (2, -1) and (8, 4).
Step 1: Calculate the slope (m)
The formula for the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Plugging in our points \( (2, -1) \) and \( (8, 4) \):
\[ m = \frac{4 - (-1)}{8 - 2} = \frac{4 + 1}{8 - 2} = \frac{5}{6} \]
Step 2: Write the equation in point-slope form
The point-slope form of a line is given by:
\[ y - y_1 = m(x - x_1) \]
Using the point (2, -1) and the slope \( m = \frac{5}{6} \):
\[ y - (-1) = \frac{5}{6}(x - 2) \]
This simplifies to:
\[ y + 1 = \frac{5}{6}(x - 2) \]
Step 3: Rewrite the equation in standard form using integers
First, we can distribute \( \frac{5}{6} \):
\[ y + 1 = \frac{5}{6}x - \frac{5}{3} \]
Next, we’ll get rid of the fractions by multiplying every term by 6:
\[ 6(y + 1) = 6\left(\frac{5}{6}x - \frac{5}{3}\right) \]
This simplifies to:
\[ 6y + 6 = 5x - 10 \]
Next, we rearrange this into standard form \( Ax + By = C \):
\[ -5x + 6y + 16 = 0 \]
To have the \( A \) term positive, we can multiply through by -1:
\[ 5x - 6y = 16 \]
Thus, the final equations are:
a. Point-slope form:
\[
y + 1 = \frac{5}{6}(x - 2)
\]
b. Standard form:
\[
5x - 6y = 16
\]