# What is the equation of the line that passes through (-1,1)  and is perpendicular to the line that passes through the following points: (13,-1),(8,4) ?

Aug 25, 2017

See a solution process below:

#### Explanation:

First, we need to find the slope of the for the two points 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:

$m = \frac{\textcolor{red}{4} - \textcolor{b l u e}{- 1}}{\textcolor{red}{8} - \textcolor{b l u e}{13}} = \frac{\textcolor{red}{4} + \textcolor{b l u e}{1}}{\textcolor{red}{8} - \textcolor{b l u e}{13}} = \frac{5}{-} 5 = - 1$

Let's call the slope for the line perpendicular to this ${m}_{p}$

The rule of perpendicular slopes is: ${m}_{p} = - \frac{1}{m}$

Substituting the slope we calculated gives:

${m}_{p} = \frac{- 1}{-} 1 = 1$

We can now use the point-slope formula to write an equation for the line. 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.

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

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

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

We can also use the slope-intercept formula. 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.

Substituting the slope we calculated gives:

$y = \textcolor{red}{1} x + \textcolor{b l u e}{b}$

We can now substitute the values from the point in the problem for $x$ and $y$ and solve for $\textcolor{b l u e}{b}$

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

$1 = - 1 + \textcolor{b l u e}{b}$

$\textcolor{red}{1} + 1 = \textcolor{red}{1} - 1 + \textcolor{b l u e}{b}$

$2 = 0 + \textcolor{b l u e}{b}$

$2 = \textcolor{b l u e}{b}$

Substituting this into the formula with the slope gives:

$y = \textcolor{red}{1} x + \textcolor{b l u e}{2}$

Aug 25, 2017

The equation of the line is $x - y = - 2$

#### Explanation:

The slope of the line passing through $\left(13 , - 1\right) \mathmr{and} \left(8 , 4\right)$ is

${m}_{1} = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}} = \frac{4 + 1}{8 - 13} = \frac{5}{-} 5 = - 1$

The product of slopes of two perpendicular lines is $m \cdot {m}_{1} = - 1$

$\therefore m = - \frac{1}{m} _ 1 = - \frac{1}{-} 1 = 1$ . So the slope of the line passing

through $\left(- 1 , 1\right)$ is $m = 1$.

The equation of the line passing through $\left(- 1 , 1\right)$ is

$y - {y}_{1} = m \left(x - {x}_{1}\right) = y - 1 = 1 \left(x + 1\right) = y - 1 = x + 1 \mathmr{and} x - y = - 2$.

The equation of the line is $x - y = - 2$ [Ans]