How do you write an equation of a line passing through (-2, -2), perpendicular to y=x+1?

Apr 14, 2017

See the entire solution process below:

Explanation:

Because the equation given in the problem is in slope-intercept form we can easily determine the slope. 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.

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

Therefore the slope of the line from the problem is: $\textcolor{red}{m = 1}$

Let's call the slope of a perpendicular line ${m}_{p}$.

The slope of a perpendicular line is: ${m}_{p} = - \frac{1}{m}$

Substituting $1$ for $m$ gives the slope of the perpendicular line in this problem as:

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

Because we are given a point we can use the point-slope formula to find 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.

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

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

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

We can also substitute the values from the problem and the slope we calculated into the slope-intercept formula and solve for $b$ to find the equation in slope-intercept form:

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

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

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

$- 4 = 0 + \textcolor{b l u e}{b}$

$- 4 = \textcolor{b l u e}{b}$

$y = \textcolor{red}{- 1} x - \textcolor{b l u e}{4}$

Solution 2) $y = \textcolor{red}{-} x - \textcolor{b l u e}{4}$