Question

How do you write an equation for the line through (6,8) and (2,-10)?

Answer 1

`y - 8 = 9/2(x - 6)` or `y = 9/2x - 19`

Explanation:

To write an equation for a line through two points we need to use a two step process.

Step 1) Find the slope.

Step 2) Using the slope, one of the points and the point-slope formula determine the equation.

The slope can be found by using the formula: `color(red)(m = (y_2 - y_1)/(x_2 - x_1)`
Where `m` is the slope and `(color(red)((x_1, y_1)))` and `(color(red)((x_2, y_2)))` are the two points.

Substituting the points we are given in this problem the slope is:

`m = (-10 - 8)/(2 - 6)`

`m = (-18)/(-4)`

`m = (-2 xx 9)/(-2 xx 2)`

`m = (-2)/(-2) xx 9/2`

`m = 1 xx 9/2`

`m = 9/2`

Now we can use one of the points we were given, the slope we determined and the point-slope formula to find the equation for the line.

The point-slope formula states: `color(red)((y - y_1) = m(x - x_1))`
Where `color(red)(m)` is the slope and `color(red)(((x_1, y_1)))` is a point the line passes through.

`y - 8 = 9/2(x - 6)`

If we want to put this into slope-intercept form we can solve for `y`:

`y - 8 = 9/2x - (9/2 xx 6)`

`y - 8 = 9/2x - (9 xx 3)`

`y - 8 = 9/2x - 27`

`y - 8 + 8 = 9/2x - 27 + 8`

`y - 0 = 9/2x - 19`

`y = 9/2x - 19`