Question

How do you write an equation of a line going through (-1,2), (3,-4)?

Answer 1

`y - 2 = -3/2(x + 1)`

or

`y = -3/2x + 1/2`

Explanation:

To find a linear equation for the line going through these two points we can use the point-slope formula.

However, first we need to determine the slope of the line.

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 two points given in the problem we can solve for `m` as:

`m = (-4 - 2)/(3 - (-1))`

`m = -6/4 = (2/2)(-3/2) = 1(-3/2)`

`m = -3/2`

Now that we have the slope we can use the slope-point formula to write 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.

Substituting the slope of `-3/2` and using the point `(-1, 2)` we can get the equation of the line as:

`y - 2 = -3/2(x - (-1))`

`y - 2 = -3/2(x + 1)`

If we want this in the slope-intercept for we can solve for `y` as follows:

`y - 2 = -3/2x - (3/2 * 1)`

`y - 2 = -3/2x - 3/2`

`y - 2 + 2 = -3/2x - 3/2 + 2`

`y - 0 = -3/2x - 3/2 + 2`

`y = -3/2x - 3/2 + 2`

`y = -3/2x - 3/2 + 2(2/2)`

`y = -3/2x - 3/2 + 4/2`

`y = -3/2x + 1/2`