Question

What is an equation for the line that passes through the coordinates (-1,2) and (7,6)?

Answer 1

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

Or

`y = 1/2x + 5/2`

Explanation:

We will use the point-slope formula to determine the line passing through these two points.

However, we will need to first calculate the slope which we can do because we have two points.

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 two points from the problem gives the result:

`m = (color(red)(6) - color(blue)(2))/(color(red)(7) - color(blue)(-1))`

`m = 4/8 = 1/2`

Now, having the slope, we can use it and either of the points in the point-slope formula to find the equation of the line we are looking for.

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.

Substituting results in:

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

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

Or, if we want to convert to the more familiar slope-intercept form we can solve for `y`:

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

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

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

`y - 0 = color(blue)(1/2)x + 1/2 + 4/2`

`y = 1/2x + 5/2`