Write an equation for the line through (-3, 2) and (6, 1).

Question:

Write an equation for the line through {eq}(-3,\ 2) {/eq} and {eq}(6,\ 1) {/eq}.

Equation of a Line:

To find the equation of a line through two given points {eq}(x_1,y_1) {/eq} and {eq}(x_2, y_2) {/eq}:

(i) Find its slope using: {eq}m= \dfrac{y_2-y_1}{x_2-x_1} {/eq}.

(ii) The equation of the line is found using: {eq}y-y_1=m(x-x_1) {/eq}.

The points on the line are:

$$(x_1,y_1)=(-3, 2)\\ (x_2, y_2)= (6,1)$$

The slope of the line is found using:

\begin{align} m&= \dfrac{y_2-y_1}{x_2-x_1} \\ &= \dfrac{1-2}{6-(-3)} \\ &= \dfrac{-1}{9} \end{align}

The equation of the line is found using:

$$y-y_1=m(x-x_1)\\ y-2=\dfrac{-1}{9} (x-(-3)) \\ y-2=\dfrac{-1}{9} (x+3) \\ y-2=\dfrac{-1}{9} x -\dfrac{1}{3}\\ \text{Adding 2 on both sides}, \\ y=\dfrac{-1}{9} x -\dfrac{1}{3}+2 \\ \boxed{\mathbf{y=\dfrac{-1}{9} x +\dfrac{5}{3}}}$$.