Question

Write an equation in point slope form of the line that passes through the two
points given. (9.-2) and (-3, 2)

Answer 1

Answer:

$\displaystyle (y + 2) = -\frac{1}{3}\, (x - 9)$.

Equivalently:

$\displaystyle (y - 2) = -\frac{1}{3}\, (x + 3)$.

Step-by-step explanation:

Let $(x_{1},\, y_{1})$ and $(x_{2}\, y_{2})$ denote the coordinates of two points in the plane. If $x_{1} \ne x_{2}$ (i.e., the two points aren't on the same vertical line,) the slope of the line that goes through the two points would be:

$\begin{aligned} m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}\end{aligned}$.

In this question, $x_{1} = 9$ and $y_{1} = (-2)$ (for $(9,\, -2)$) while $x_{2} = (-3)$ and $y_{2} = 2$ (for $(-3,\, 2)$.) Given that $x_{1} \ne x_{2}$, the slope of the line that goes through these two points would be:

$\begin{aligned} m &= \frac{y_{2} - y_{1}}{x_{2} - x_{1}} \\ &= \frac{2 - (-2)}{(-3) - 9} \\ &= -\frac{1}{3}\end{aligned}$

If a line in a plane is of slope $m$ and goes through the point $(x_{0},\, y_{0})$, the point-slope equation of this line would be:

$(y - y_{0}) = m\, (x - x_{0})$.

The slope of this line is $m = (-1/3)$. If the point $(9,\, -2)$ is chosen as $(x_{0},\, y_{0})$ ($x_{0} = 9$ and $y_{0} = (-2)$,) then the point-slope equation of this line would be:

$(y - (-2)) = (-1/3)\, (x - 9)$.

Simplify to obtain:

$\displaystyle (y + 2) = -\frac{1}{3}\, (x - 9)$.

Similarly, if the point $(-3,\, 2)$ is chosen as $(x_{0},\, y_{0})$, the point-slope equation of this would be:

$\displaystyle (y - 2) = -\frac{1}{3}\, (x + 3)$.