Q10. - Algebra 2

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 $\mathbf{y}=\mathbf{x}-\mathbf{2}$

Step-by-Step Solution

Answer

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

Step 1 – 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 $y-k=m\left(x-h\right)$.

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

Step 3 – 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, $m=-1$.

Step 4 – 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 $y=-x-2$.