# What is the slope-intercept form of the line passing through  (5, 1) and  (0, -6) ?

Mar 3, 2018

The general slope intercept form of a line is

$y = m x + c$

where $m$ is the slope of the line and $c$ is its $y$-intercept (the point at which the line cuts the $y$ axis).

#### Explanation:

First, get all the terms of the equation. Let us calculate the slope.

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

$= \frac{- 6 - 1}{0 - 5}$

$= \frac{7}{5}$

The $y$-intercept of the line is already given. It is $- 6$ since the $x$ coordinate of the line is zero when it intersects the $y$ axis.

$c = - 6$

Use the equation.

$y = \left(\frac{7}{5}\right) x - 6$

$y = 1.4 x + 6$

#### Explanation:

$P \equiv \left(5 , 1\right)$
$Q \equiv \left(0 , - 6\right)$
$m = \frac{- 6 - 1}{0 - 5} = - \frac{7}{-} 5$
$m = 1.4$
$c = 1 - 1.4 \times 5 = 1 - 7$
$c = 6$
$y = m x + c$
$y = 1.4 x + 6$

Mar 3, 2018

One answer is: $\left(y - 1\right) = \frac{7}{5} \left(x - 5\right)$
the other is: $\left(y + 6\right) = \frac{7}{5} \left(x - 0\right)$

#### Explanation:

The slope-intercept form of a line tells you what you need to find first: the slope.
Find slope using $m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$
where $\left({x}_{1} , {y}_{1}\right)$ and $\left({x}_{2} , {y}_{2}\right)$ are the given two points
$\left(5 , 1\right)$ and $\left(0 , - 6\right)$:

$m = \frac{- 6 - 1}{0 - 5} = \frac{- 7}{-} 5 = \frac{7}{5}$

You can see this is in both answers.

Now choose either point and plug in to the slope-intercept form of a line: $\left(y - {y}_{1}\right) = m \left(x - {x}_{1}\right)$

Choosing the first point results in the first answer and choosing the second point yields the second answer. Also note that the second point is technically the y-intercept, so you could write the equation in slope-intercept form ($y = m x + b$): $y = \frac{7}{5} x - 6$.