# How do you write the equation of a line in slope intercept, point slope and standard form given (-1,2) and (4,-1)?

Aug 28, 2017

See a 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:

$m = \frac{\textcolor{red}{- 1} - \textcolor{b l u e}{2}}{\textcolor{red}{4} - \textcolor{b l u e}{- 1}} = \frac{\textcolor{red}{- 1} - \textcolor{b l u e}{2}}{\textcolor{red}{4} + \textcolor{b l u e}{1}} = - \frac{3}{5}$

We can now use the slope we calculated and the values from the first point in the problem to write an equation in point-slope form. The point-slope form of a linear equation is: $\left(y - \textcolor{b l u e}{{y}_{1}}\right) = \textcolor{red}{m} \left(x - \textcolor{b l u e}{{x}_{1}}\right)$

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

$\left(y - \textcolor{b l u e}{2}\right) = \textcolor{red}{- \frac{3}{5}} \left(x - \textcolor{b l u e}{- 1}\right)$

$\left(y - \textcolor{b l u e}{2}\right) = \textcolor{red}{- \frac{3}{5}} \left(x + \textcolor{b l u e}{1}\right)$

We can also use the slope we calculated and the values from the second point in the problem to write an equation in point-slope form.

$\left(y - \textcolor{b l u e}{- 1}\right) = \textcolor{red}{- \frac{3}{5}} \left(x - \textcolor{b l u e}{4}\right)$

$\left(y + \textcolor{b l u e}{1}\right) = \textcolor{red}{- \frac{3}{5}} \left(x - \textcolor{b l u e}{4}\right)$

We can solve this equation for $y$ to write an equation is slope-intercept form. The slope-intercept form of a linear equation is: $y = \textcolor{red}{m} x + \textcolor{b l u e}{b}$

Where $\textcolor{red}{m}$ is the slope and $\textcolor{b l u e}{b}$ is the y-intercept value.

$y + \textcolor{b l u e}{1} = - \frac{3}{5} x - \left(- \frac{3}{5} \times \textcolor{b l u e}{4}\right)$

$y + \textcolor{b l u e}{1} = - \frac{3}{5} x + \frac{12}{5}$

$y + \textcolor{b l u e}{1} - 1 = - \frac{3}{5} x + \frac{12}{5} - 1$

$y + 0 = - \frac{3}{5} x + \frac{12}{5} - \frac{5}{5}$

$y = \textcolor{red}{- \frac{3}{5}} x + \textcolor{b l u e}{\frac{7}{5}}$

We can now convert this equation to Standard Linear form. 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

$\frac{3}{5} x + y = \frac{3}{5} x + \textcolor{red}{- \frac{3}{5}} x + \textcolor{b l u e}{\frac{7}{5}}$

$\frac{3}{5} x + y = 0 + \textcolor{b l u e}{\frac{7}{5}}$

$\frac{3}{5} x + y = \textcolor{b l u e}{\frac{7}{5}}$

$5 \left(\frac{3}{5} x + y\right) = 5 \times \textcolor{b l u e}{\frac{7}{5}}$

$\textcolor{red}{3} x + \textcolor{b l u e}{5} y = \textcolor{g r e e n}{7}$