# How do you write an equation of a line through (-3,-1) and (1/2,2)?

##### 1 Answer
Jun 13, 2017

The equation of that line is $y = \frac{7 x + 15}{6}$.

#### Explanation:

The equation of a line is based on two simple questions: "How much $y$ changes when you add $1$ to $x$?" and "How much is $y$ when $x = 0$?"

First, it's important to know that a linear equation has a general formula defined by $y = m \cdot x + n$.

Having those questions in mind, we can find the slope ($m$) of the line, that is how much $y$ changes when you add $1$ to $x$:
$m = \frac{{D}_{x}}{{D}_{y}}$, with ${D}_{x}$ being the difference in $x$ and ${D}_{y}$ being the difference in $y$.

${D}_{x} = - 3 - \left(\frac{1}{2}\right) = - 3 - \frac{1}{2} = - \frac{7}{2}$
${D}_{y} = - 1 - \left(2\right) = - 1 - 2 = - 3$

$m = \frac{- \frac{7}{2}}{-} 3 = - \frac{7}{2} \cdot - \frac{1}{3} = \frac{- 7 \cdot - 1}{2 \cdot 3} = \frac{7}{6}$

Now, we need to find ${y}_{0}$, that is the value of $y$ when $x = 0$:
Since when $x = - 3$, $y = - 1$, we can sum the slope $3$ times (since $- 3 + 3 = 0$) in $y$:

${y}_{0} = - 1 + 3 \cdot \left(\frac{7}{6}\right) = - 1 + \frac{21}{6} = - \frac{6}{6} + \frac{21}{6} = \frac{15}{6}$

We now have the slope and the ${y}_{0}$ (or $n$) value, we apply in the main formula of a linear equation:

$y = m \cdot x + n = \frac{7}{6} \cdot x + \frac{15}{6} = \frac{7 x + 15}{6}$