# How do you write the standard form of the equation given (3,1) and slope 1/2?

Feb 26, 2017

$\textcolor{red}{1} x - \textcolor{b l u e}{2} y = \textcolor{g r e e n}{1}$

Or

$x - \textcolor{b l u e}{2} y = \textcolor{g r e e n}{1}$

#### Explanation:

First, we can use the point-slope formula to find 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 values from the problem gives:

$\left(y - \textcolor{red}{1}\right) = \textcolor{b l u e}{\frac{1}{2}} \left(x - \textcolor{red}{3}\right)$

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

We can transform our equation into the standard form as follows:

First, multiply each side of the equation by $\textcolor{red}{2}$ to eliminate all fractions while keeping the equation balanced:

$\textcolor{red}{2} \left(y - \textcolor{red}{1}\right) = \textcolor{red}{2} \times \textcolor{b l u e}{\frac{1}{2}} \left(x - \textcolor{red}{3}\right)$

$\left(\textcolor{red}{2} \times y\right) - \left(\textcolor{red}{2} \times \textcolor{red}{1}\right) = x - \textcolor{red}{3}$

$2 y - 2 = x - 3$

Next, subtract $\textcolor{red}{x}$ and add $\textcolor{b l u e}{2}$ to each side of the equation to put the $x$ and $y$ terms on the left side of the equation while keeping the equation balanced:

$- \textcolor{red}{x} + 2 y - 2 + \textcolor{b l u e}{2} = - \textcolor{red}{x} + x - 3 + \textcolor{b l u e}{2}$

$- x + 2 y - 0 = 0 - 1$

$- x + 2 y = - 1$

Now, multiply each side of the equation by $\textcolor{red}{- 1}$ to convert the $x$ coefficient to a positive integer while keeping the equation balanced:

$\textcolor{red}{- 1} \left(- x + 2 y\right) = \textcolor{red}{- 1} \times - 1$

$\left(\textcolor{red}{- 1} \times - x\right) + \left(\textcolor{red}{- 1} \times 2 y\right) = 1$

$\textcolor{red}{1} x - \textcolor{b l u e}{2} y = \textcolor{g r e e n}{1}$