# What is the equation of the line between (3,-13) and (-7,1)?

May 30, 2018

$y = - \setminus \frac{7}{5} x - \frac{44}{5}$

#### Explanation:

When you know the coordinates of two points ${P}_{1} = \left({x}_{1} , {y}_{1}\right)$ and ${P}_{2} = \left({x}_{2} , {y}_{2}\right)$, the line passing through them has equation

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

$\setminus \frac{y + 13}{1 + 13} = \setminus \frac{x - 3}{- 7 - 3} \setminus \iff \setminus \frac{y + 13}{14} = \setminus \frac{x - 3}{- 10}$

Multiply both sides by $14$:

$y + 13 = - \setminus \frac{7}{5} x + \setminus \frac{42}{10}$

Subtract $13$ from both sides:

$y = - \setminus \frac{7}{5} x - \frac{44}{5}$

May 30, 2018

Over the top detail given so that you can see where everything comes from.

$y = - \frac{7}{5} x - \frac{44}{5}$

#### Explanation:

Reading left to right on the x-axis.
Set point 1 as ${P}_{1} \to \left({x}_{1} , {y}_{1}\right) = \left(- 7 , 1\right)$
Set point 2 as ${P}_{2} \to \left({x}_{2} , {y}_{2}\right) = \left(3 , - 13\right)$

In reading this we 'travel' from ${x}_{1}$ to ${x}_{2}$ so to determine the difference we have ${x}_{2} - {x}_{1} \mathmr{and} {y}_{2} - {y}_{1}$

$\textcolor{red}{m} = \left(\text{change in y")/("change in x}\right) \to \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}} = \frac{- 13 - 1}{3 - \left(- 7\right)} = \textcolor{red}{\frac{- 14}{+ 10} = - \frac{7}{5}}$

We may choose any of the two: ${P}_{1} \text{ or } {P}_{2}$ for the next bit. I choose ${P}_{1}$

$m = - \frac{7}{5} = \frac{{y}_{2} - 1}{{x}_{2} - \left(- 7\right)} = \frac{{y}_{2} - 1}{{x}_{2} + 7}$

$- 7 \left({x}_{2} + 7\right) = 5 \left({y}_{2} - 1\right)$

$- 7 {x}_{2} - 49 = 5 {y}_{2} - 5$

$- 7 {x}_{2} - 44 = 5 {y}_{2}$

Divide both sides by 5

$- \frac{7}{5} {x}_{2} - \frac{44}{5} = {y}_{2}$

Now using generic $x \mathmr{and} y$

$- \frac{7}{5} x - \frac{44}{5} = y$

$y = - \frac{7}{5} x - \frac{44}{5}$