Question

How do you write an equation of a line passing through (4,8) and (1,2)?

Answer 1

`(y - color(red)(8)) = color(blue)(2)(x - color(red)(4))`

Or

`(y - color(red)(2)) = color(blue)(2)(x - color(red)(1))`

Explanation:

We can use the point-slope formula to write the equation of a line passing through these two points. First, however, we must determine the slope.

The slope can be found by using the formula: `m = (color(red)(y_2) - color(blue)(y_1))/(color(red)(x_2) - color(blue)(x_1))`

Where `m` is the slope and (`color(blue)(x_1, y_1)`) and (`color(red)(x_2, y_2)`) are the two points on the line.

Substituting the values from the points in the problem gives:

`m = (color(red)(2) - color(blue)(8))/(color(red)(1) - color(blue)(4))`

`m = (-6)/(-3)`

`m = 2`

Now we can use the slope and the first point in the point-slope formula to find an equation.

The point-slope formula states: `(y - color(red)(y_1)) = color(blue)(m)(x - color(red)(x_1))`

Where `color(blue)(m)` is the slope and `color(red)(((x_1, y_1)))` is a point the line passes through.

`(y - color(red)(8)) = color(blue)(2)(x - color(red)(4))`

We can also use the slope and the second point in the point-slope formula to find an equation.

`(y - color(red)(2)) = color(blue)(2)(x - color(red)(1))`