# How do you write the equation of the line that passes through (2, 4) and (1, –3) in standard form?

Jan 16, 2017

$\textcolor{red}{7} x - \textcolor{b l u e}{1} y = \textcolor{g r e e n}{10}$

#### Explanation:

We will use the point-slope formula to first define the equation. First however we need to first use the two points from the problem to find the slope:

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 two points from the problem gives the slope as:

$m = \frac{\textcolor{red}{- 3} - \textcolor{b l u e}{4}}{\textcolor{red}{1} - \textcolor{b l u e}{2}}$

$m = - \frac{7}{-} 1 = 7$

Now that we have the slope we can use it and one of the points in the point-slope formula to get an equation for the 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.

$\left(y - \textcolor{red}{- 3}\right) = \textcolor{b l u e}{7} \left(x - \textcolor{red}{1}\right)$

$\left(y + \textcolor{red}{3}\right) = \textcolor{b l u e}{7} \left(x - \textcolor{red}{1}\right)$

We can now transform this into the standard form for a linear equation by doing the necessary mathematics.

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

$y + \textcolor{red}{3} = \left(\textcolor{b l u e}{7} \times x\right) - \left(\textcolor{b l u e}{7} \times \textcolor{red}{1}\right)$

$y + 3 = 7 x - 7$

$y + 3 - \textcolor{red}{7 x} - \textcolor{b l u e}{3} = 7 x - 7 - \textcolor{red}{7 x} - \textcolor{b l u e}{3}$

$- \textcolor{red}{7 x} + y + 3 - \textcolor{b l u e}{3} = 7 x - \textcolor{red}{7 x} - 7 - \textcolor{b l u e}{3}$

$- 7 x + y + 0 = 0 - 7 - 3$

$- 7 x + y = - 10$

$- 1 \left(- 7 x + y\right) = - 1 \times - 10$

$7 x - y = 10$

$\textcolor{red}{7} x - \textcolor{b l u e}{1} y = \textcolor{g r e e n}{10}$