Question

Write an equation in slope-intercept form for the line that satisfies each set of conditions.

Passes through $\left(\mathbf{0},-\mathbf{2}\right)$, perpendicular to the graph of $\mathbf{y}=\mathbf{x}-\mathbf{2}$

Short answer

The equation of the required straight line in slope-intercept form is $y=-x-2$.

Step-by-step solution

– State the concept 

The slope intercept form of a straight-line equation is $y=mx+c$ where $m$ is the slope and $c$ is the y-intercept.

The slope of a line perpendicular to a line having slope $m$, is $-\frac{1}{m}$.

The equation of a straight-line having slope $m$ and passing through the point $\left(h,k\right)$ is given as $y-k=m\left(x-h\right)$.

– List the given data

It is given that the line passes through $\left(0,-2\right)$ and is perpendicular to the graph of $y=x-2$.

Then, $\left(h,k\right)=\left(0,-2\right)$.

– Find the slope

Comparing the equation $y=x-2$ to the general equation of a line in slope-intercept form, $y=mx+c$, $m=1$.

This implies that the slope of $y=x-2$is $1$. Then, the slope of the required line which is perpendicular to $y=x-2$is $-1$.

Then, $m=-1$.

– Write the equation

Put $m=-1$ and $\left(h,k\right)=\left(0,-2\right)$ in $y-k=m\left(x-h\right)$ to get,

$y-\left(-2\right)=-1\left(x-0\right)$

$y+2=-x$ (Simplify)

$y+2-2=-x-2$ (Subtract 2 from both sides)

$y=-x-2$ (Simplify)

So, the required equation of the straight line in slope-intercept form is $y=-x-2$.