# How do you write the equation given (0,0); (-2,7)?

Jul 23, 2017

See a solution process below:

#### Explanation:

First, we must determine the slope of the line running through the two points given in the problem. 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:

$m = \frac{\textcolor{red}{7} - \textcolor{b l u e}{0}}{\textcolor{red}{- 2} - \textcolor{b l u e}{0}} = - \frac{7}{2}$

Now, we can use the point-slope formula to write 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 $\left(\textcolor{red}{{x}_{1} , {y}_{1}}\right)$ is a point the line passes through.

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

$\left(y - \textcolor{red}{0}\right) = \textcolor{b l u e}{- \frac{7}{2}} \left(x - \textcolor{red}{0}\right)$

$y = \textcolor{b l u e}{- \frac{7}{2}} x$

We can also substitute the slope we calculated and the values from the second point in the problem giving:

$\left(y - \textcolor{red}{7}\right) = \textcolor{b l u e}{- \frac{7}{2}} \left(x - \textcolor{red}{- 2}\right)$

$\left(y - \textcolor{red}{7}\right) = \textcolor{b l u e}{- \frac{7}{2}} \left(x + \textcolor{red}{2}\right)$

We can also take the first equation and write this in 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.

$y = \textcolor{red}{- \frac{7}{2}} x + \textcolor{b l u e}{0}$

Jul 23, 2017

$y = - \frac{7}{2} x$

#### Explanation:

The find the equation of a line or gradient, first find the slope via the formula: $m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$

If we let $\left(0 , 0\right) \to \left(\textcolor{red}{{x}_{1}} , \textcolor{b l u e}{{y}_{1}}\right)$ and (-2,7)->color(red)(x_2),color(blue)(y_2)) then,

$m = \frac{\textcolor{b l u e}{7 - 0}}{\textcolor{red}{- 2 - 0}} = - \frac{7}{2} \leftarrow$ This is the slope of the line/gradient

Now that we have found the slope we can find the equation via the point-slope formula:

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

by substituting $- \frac{7}{2}$ for $m$ and any of the two coordinates given. I will use $\left(0 , 0\right)$ be make things easier. We let $\left(0 , 0\right) \to \left({x}_{1} , {y}_{1}\right)$

Thus,

$y - 0 = - \frac{7}{2} \left(x - 0\right)$

If we simplify this, we simply get

$y = - \frac{7}{2} x \leftarrow$ This is our equation

Graph:

graph{-7/2x [-10, 10, -5, 5]}