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

`\displaystyle (y + 2) = -(1)/(3)\, (x - 9)`.

Equivalently:

`\displaystyle (y - 2) = -(1)/(3)\, (x + 3)`.

Explanation:

Let
`(x_(1),\, y_(1))` and
`(x_(2)\, y_(2))` denote the coordinates of two points in the plane. If
`x_(1) \\e 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 = (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) \\e x_(2)`, the slope of the line that goes through these two points would be:

`\begin{aligned} m &= (y_(2) - y_(1))/(x_(2) - x_(1)) \\ &= (2 - (-2))/((-3) - 9) \\ &= -(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) = -(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) = -(1)/(3)\, (x + 3)`.