# How do you write an equation of a line with points (9,-2), (4,3)?

Jan 29, 2017

$\left(y + \textcolor{red}{2}\right) = \textcolor{b l u e}{- 1} \left(x - \textcolor{red}{9}\right)$

Or

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

Or

$y = - 1 x + 7$ or $y = - x + 7$

#### Explanation:

We can use the point-slope formula to find a line passing through these two points.

First, however, we must use the two points to determine 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 values from the points given in the problem produces:

$m = \frac{\textcolor{red}{3} - \textcolor{b l u e}{- 2}}{\textcolor{red}{4} - \textcolor{b l u e}{9}}$

$m = \frac{\textcolor{red}{3} + \textcolor{b l u e}{2}}{\textcolor{red}{4} - \textcolor{b l u e}{9}}$

$m = \frac{5}{-} 5 = - 1$

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.

We can now substitute the first point from the problem and the slope we calculated to obtain an equation:

$\left(y - \textcolor{red}{- 2}\right) = \textcolor{b l u e}{- 1} \left(x - \textcolor{red}{9}\right)$

$\left(y + \textcolor{red}{2}\right) = \textcolor{b l u e}{- 1} \left(x - \textcolor{red}{9}\right)$

We also substitute the second point from the problem and the slope we calculated to obtain another equation:

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

We can solve this problem for $y$ to obtain the equation in the familiar slope-intercept form:

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

$y - \textcolor{red}{3} = - 1 x + 4$

$y - \textcolor{red}{3} + 3 = - 1 x + 4 + 3$

$y - 0 = - 1 x + 7$

$y = - 1 x + 7$ or $y = - x + 7$