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

Nov 12, 2015

$y = x - 5$

#### Explanation:

If you know that a line passes through two points, then that line is unique. If the points are $\left({x}_{1} , {y}_{1}\right)$ and $\left({x}_{2} , {y}_{2}\right)$, then the equation for the line is

$\setminus \frac{x - {x}_{2}}{{x}_{1} - {x}_{2}} = \setminus \frac{y - {y}_{2}}{{y}_{1} - {y}_{2}}$

In your case, we have $\left({x}_{1} , {y}_{1}\right) = \left(1 , - 4\right)$ and $\left({x}_{2} , {y}_{2}\right) = \left(4 , - 1\right)$

Plugging these values into the formula gives

$\setminus \frac{x - 4}{1 - 4} = \setminus \frac{y - \left(- 1\right)}{- 4 - \left(- 1\right)}$

which becomes

$\setminus \frac{x - 4}{\cancel{- 3}} = \setminus \frac{y + 1}{\cancel{- 3}}$

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

${y}_{2} = - 1 = {x}_{2} - 5 = 4 - 5$