Question

What is the equation of the line passing through `(1,-4)` and `(4,-1)`?

Answer 1

`y=x-5`

Explanation:

If you know that a line passes through two points, then that line is unique. If the points are `(x_1,y_1)` and `(x_2,y_2)`, then the equation for the line is

`\frac{x-x_2}{x_1-x_2} = \frac{y-y_2}{y_1-y_2}`

In your case, we have `(x_1,y_1)=(1,-4)` and `(x_2,y_2)=(4,-1)`

Plugging these values into the formula gives

`\frac{x-4}{1-4} = \frac{y-(-1)}{-4-(-1)}`

which becomes

`\frac{x-4}{cancel(-3)} = \frac{y+1}{cancel(-3)}`

Isolating the `y` term, we arrive at the form `y=x-5`

Let's verify:
our two points satisfy this equation, because the `y` coordinate is smaller than the `x` coordinate by `5` units:

`y_1=-4 = x_1-5=1-5`, and

`y_2 = -1 = x_2-5 = 4-5`