Beginning Algebra With Applications, Chapter 5, 5.4, Section 5.4, Problem 64

Is the linear equation that contains all the ordered pairs $(3,-1),(12,-4),(-6,2)$? If there is, find the equation.

To check if the given pairs has a linear equation, we find first the slope. So we let $(x_1, y_1) = (3,-1)$ and $(x_2, y_2) = (12,-4)$,

$\displaystyle m = \frac{-4-(-1)}{12-3} = \frac{-4+1}{9} = \frac{-3}{9} = \frac{-1}{3}$

Then we let $(x_1, y_1) = (12,-4)$ and $(x_2, y_2) = (-6,2)$

$\displaystyle m = \frac{2-(-4)}{-6-12}= \frac{2+4}{-18} = \frac{-6}{18} = \frac{-1}{3}$

And we let $(x_1, y_1) = (3, -1)$ and $(x_2, y_2) = (-6,2)$

$\displaystyle m = \frac{2-(-1)}{-6-3} = \frac{2+1}{-9} = \frac{-3}{9} = \frac{-1}{3}$

Since they have the same slope. The given ordered pairs has a linear equation

Using the point-slope formula, where $\displaystyle m = \frac{-1}{3}$ and $(x_1, y_1) = (3,-1)$


$
\begin{equation}
\begin{aligned}

y - y_1 =& m(x -x_1)
\\
\\
y - (-1) =& \frac{-1}{3} (x-3)
\\
\\
y + 1 =& \frac{-1}{3}x + 1
\\
\\
y =& \frac{-1}{3}x

\end{aligned}
\end{equation}
$


The equation is $\displaystyle y = \frac{-1}{3}x$.