# How do you write in standard form an equation of the line with the slope -4 through the given point (2,2)?

May 11, 2018

y $=$ -4x $+$ 10

#### Explanation:

First off you have to know the standard form formula which is:
y $=$ mx $+$ b

Plug in the slope and the points (x,y) to get b

y $=$ mx $+$ b
2 $=$ -4(2) $+$ b
2 $=$ -8 $+$ b

Next, you add 8 to both sides to get b alone:
10 $=$ b

Plug your slope and b value into the standard formula

May 11, 2018

$4 x + y = 10$

#### Explanation:

Let's start with the very definition of the slope of a line: take two points ${P}_{1} = \left({x}_{1} , {y}_{1}\right)$ and ${P}_{2} = \left({x}_{2} , {y}_{2}\right)$, the slope $m$ is defined as

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

From here, we have

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

To have the generic expression of the line, let's change this equation a little bit. Instead of having two fixed points ${P}_{1} = \left({x}_{1} , {y}_{1}\right)$ and ${P}_{2} = \left({x}_{2} , {y}_{2}\right)$, assume we have a fixed point ${P}_{0} = \left({x}_{0} , {y}_{0}\right)$ and any other generic point on the line $P = \left(x , y\right)$. The equation becomes

$y - {y}_{0} = m \left(x - {x}_{0}\right)$

Plug your values: ${P}_{0} = \left(2 , 2\right)$, and $m = - 4$, to get

$y - 2 = - 4 \left(x - 2\right)$

From here, with a bit of algebra you get

$y = - 4 x + 10$

EDIT:

as pointed out, we're not in the standard form yet. To achieve it, we must separate variables from "pure" numbers. Just move $- 4 x$ to the left hand side to obtain

$4 x + y = 10$

May 11, 2018

$4 x + y = 10$

#### Explanation:

$\text{the equation of a line in "color(blue)"standard form}$ is.

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{A x + B y = C} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

$\text{where A is a positive integer and B, C are integers}$

$\text{obtain the equation in "color(blue)"point-slope form ""and}$
$\text{rearrange into standard form}$

•color(white)(x)y-y_1=m(x-x_1)

$\text{where m is the slope and "(x_1,y_1)" a point on the line}$

$\text{here "m=-4" and } \left({x}_{1} , {y}_{1}\right) = \left(2 , 2\right)$

$\Rightarrow y - 2 = - 4 \left(x - 2\right) \leftarrow \textcolor{b l u e}{\text{in point-slope form}}$

$\Rightarrow y - 2 = - 4 x + 8$

$\text{add "4x" to both sides}$

$\Rightarrow 4 x + y - 2 = 8$

$\text{add 2 to both sides}$

$\Rightarrow 4 x + y = 10 \leftarrow \textcolor{red}{\text{in standard form}}$