Question

A line passes through the points (7,9) and (10,1). What is its equation in point-slope form?

Use one of the specified points in your equation. Write your answer using integers, proper fractions, and improper fractions. Simplify all fractions.

Answer 1

`(y - 9) = (-8/3)\, (x - 7)`.

Explanation:

If a line in a cartesian plane has slope
`m`, and the point
`(x_(0),\, y_(0))` is on this line, then the point-slope equation of this line will be
`(y - y_(0)) = m\, (x - x_(0))`.

The slope of a line measures the rate of change in
`y`-coordinates relative to the change in the
`x`-coordinates. If a line goes through two points
`(x_(0),\, y_(0))` and
`(x_(1),\, y_(1))`, the slope of that line will be:

`\begin{aligned}m &= (y_(1) - y_(0))/(x_(1) - x_(0))\end{aligned}`.

In this question, the two points on this line are
`(7,\, 9)` and
`(10,\, 1)`, such that
`x_(0) = 7`,
`y_(0) = 9`,
`x_(1) = 10`, and
`y_(1) = 1`. Substitute these values into the equation to find the slope of this line:

`\begin{aligned}m &= (y_(1) - y_(0))/(x_(1) - x_(0)) \\ &= (1 - 9)/(10 - 7) \\ &= \left(-(8)/(3)\right)\end{aligned}`.

With the point
`(7,\, 9)` as the specific point
`(x_(0),\, y_(0))` (such that
`x_(0) = 7` and
`y_(0) = 1`) as well as a slope of
`m = (-8 / 3)`, the point-slope equation of this line will be:

`y - y_(0) = m\, (x - x_(0))`.

`\displaystyle y - 9 = \left(-(8)/(3)\right)\, (x - 7)`.