Question

A line passes through (2, –1) and (8, 4).Write an equation for the line in point-slope form.
Rewrite the equation in standard form using integers.


y – 2 = (x + 1); –5x + 6y = 17


y + 1 = (x – 2); –5x + 6y = –16


y + 1 = (x + 2); –5x + 6y = –16


y – 1 = (x – 2); –5x + 6y = 16

Answer 1

To find the equation of the line that passes through the points (2, –1) and (8, 4) in point-slope form, we first need to calculate the slope (m) of the line.

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

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

Substituting the coordinates of the points (2, -1) and (8, 4):

\[
m = \frac{4 - (-1)}{8 - 2} = \frac{4 + 1}{8 - 2} = \frac{5}{6}
\]

Now that we have the slope \( m = \frac{5}{6} \), we can use the point-slope form of the equation of a line, which is:

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

Using point (2, -1):

\[
y - (-1) = \frac{5}{6}(x - 2)
\]

This simplifies to:

\[
y + 1 = \frac{5}{6}(x - 2)
\]

Now we will convert this to standard form \( Ax + By = C \), which requires removing the fraction and rearranging the equation.

Starting with:

\[
y + 1 = \frac{5}{6}(x - 2)
\]

Multiply every term by 6 to eliminate the fraction:

\[
6(y + 1) = 5(x - 2)
\]

Expanding both sides:

\[
6y + 6 = 5x - 10
\]

Rearranging gives:

\[
-5x + 6y + 6 = -10
\]

Adding 10 to both sides:

\[
-5x + 6y + 16 = 0
\]

Therefore, we can rearrange this into standard form, typically in the form \( Ax + By = C \):

\[
-5x + 6y = -16
\]

To make A positive, we can multiply the entire equation by -1:

\[
5x - 6y = 16
\]

So the equation of the line in standard form is:

\[
5x - 6y = 16
\]

Thus, the answer that matches with your choices is:

**y + 1 = (x – 2); –5x + 6y = 16**.