# How do you write an equation in standard form given point (-2,4) and (0,6)?

Apr 2, 2017

See the entire solution process below:

#### Explanation:

First, we need to determine the slope of the line. The slope can be found by using the formula: $m = \frac{\textcolor{red}{{y}_{2}} - \textcolor{b l u e}{{y}_{1}}}{\textcolor{red}{{x}_{2}} - \textcolor{b l u e}{{x}_{1}}}$

Where $m$ is the slope and ($\textcolor{b l u e}{{x}_{1} , {y}_{1}}$) and ($\textcolor{red}{{x}_{2} , {y}_{2}}$) are the two points on the line.

Substituting the values from the points in the problem gives:

$m = \frac{\textcolor{red}{6} - \textcolor{b l u e}{4}}{\textcolor{red}{0} - \textcolor{b l u e}{- 2}} = \frac{\textcolor{red}{6} - \textcolor{b l u e}{4}}{\textcolor{red}{0} + \textcolor{b l u e}{2}} = \frac{2}{2} = 1$

We can now use the point-slope formula to write an equation for this line. The point-slope formula states: $\left(y - \textcolor{red}{{y}_{1}}\right) = \textcolor{b l u e}{m} \left(x - \textcolor{red}{{x}_{1}}\right)$

Where $\textcolor{b l u e}{m}$ is the slope and $\textcolor{red}{\left(\left({x}_{1} , {y}_{1}\right)\right)}$ is a point the line passes through.

Substituting the slope we calculated and the values from the second point in the problem gives:

$\left(y - \textcolor{red}{6}\right) = \textcolor{b l u e}{1} \left(x - \textcolor{red}{0}\right)$

$y - \textcolor{red}{6} = 1 x$

The standard form of a linear equation is: $\textcolor{red}{A} x + \textcolor{b l u e}{B} y = \textcolor{g r e e n}{C}$

Where, if at all possible, $\textcolor{red}{A}$, $\textcolor{b l u e}{B}$, and $\textcolor{g r e e n}{C}$are integers, and A is non-negative, and, A, B, and C have no common factors other than 1

To start transforming the equation we wrote to standard form we can add $\textcolor{red}{6}$ and subtract $\textcolor{b l u e}{1 x}$ from each side of the equation to put both variables on the left side of the equation and the constant on the right side of the equation:

$- \textcolor{b l u e}{1 x} + y - \textcolor{red}{6} + \textcolor{red}{6} = - \textcolor{b l u e}{1 x} + 1 x + \textcolor{red}{6}$

$- 1 x + y - 0 = 0 + 6$

$- 1 x + y = 6$

Now, because the coefficient of the $x$ variable should be a positive integer we can multiply each side of the equation by $\textcolor{red}{- 1}$ to change the sign of the $x$ variable while keeping the equation balanced:

$\textcolor{red}{- 1} \left(- 1 x + y\right) = \textcolor{red}{- 1} \times 6$

$\left(\textcolor{red}{- 1} \times - 1 x\right) + \left(\textcolor{red}{- 1} \times y\right) = - 6$

$\textcolor{red}{1} x + \textcolor{b l u e}{- 1} y = \textcolor{g r e e n}{- 6}$