Question

Is each line parallel, perpendicular, or neither parallel nor perpendicular to the line x + 2y = 6?

Answer 1

ANSWER

We need to rewrite $x+2y=6$, in slope intercept form.

$\Rightarrow 2y=-x+6$,

$\Rightarrow y=-\frac{1}{2}x+3$

The given equation has a slope of $-\frac{1}{2}$.

ANSWER TO QUESTION 1

We now the write given options also in slope intercept form.

For the first one, we have

$y=-\frac{1}{2}x+5$

This first equation also has a slope  of $-\frac{1}{2}$.

Hence $y=-\frac{1}{2}x+5$ is parallel to $x+2y=6$.

ANSWER TO QUESTION 2:

The second equation is;

$-2x+y=-4$.

We need to write this one too in slope intercept form.

$y=2x-4$

This second equation has a slope of 2

Since $2\times -\frac{1}{2}=-1$, the line $-2x+y=-4$

is perpendicular to $x+2y=6$.

ANSWER TO QUESTION 3

We rewrite the equation $-x+2y=2$ in slope intercept form.

$\Rightarrow 2y=x+2$

$\Rightarrow y=\frac{1}{2}x+1$

This equation also has a slope of $\frac{1}{2}[\tex]since the slope of [tex]-x+2y=2$ is not equal to the slope of $x+2y=6$, and the product of these two slopes $\frac{1}{2} \times - \frac{1}{2} \ne -1$, the two lines are neither parallel nor perpendicular.

QUESTION 4

The given line has equation $x+2y=-2$. We rewrite this equation in the slope intercept form.

$\Rightarrow 2y=-x-2$

$\Rightarrow y=-\frac{1}{2}x-1$

This line has a slope of $-\frac{1}{2}$.

Since this slope is the same as the slope of the given line,

$y=-\frac{1}{2}x-1$ is parallel to  $x+2y=6$.

See graph in attachment.

Answer 2

Y=-1/2x+5 is parallel.
-2x+y=-4 is perpendicular.
-x+2y=2 is neither.
x+2y=2 is parallel.
Hope this helps