When comparing two lines to see if they are parallel, they must have the same slope.  To find the slope of a line, we write it in slope-intercept form

$\displaystyle y = mx + b$

where m is the slope.  

The original equation

$\displaystyle 4y = -2x + 8$

will need to be written in slope-intercept form.  To do that, we will divide each term by 4

$\displaystyle \frac{4y}{4} = \frac{-2x}{4} + \frac{8}{4}$

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

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

Therefore, the slope of the original line is $\displaystyle -\frac{1}{2}$.  A line that is parallel to this line will also have a slope of $\displaystyle -\frac{1}{2}$.

Therefore, the line

$\displaystyle y = -\frac{1}{2}x -4$

is parallel to the original line.

If two lines are parallel, then they have the same slope.  To find the slope of a line, we write it in slope-intercept form

$\displaystyle y = mx+b$

where m is the slope.  So given the equation

$\displaystyle 5y = -15x + 20$

we must solve for y.  To do that, we will divide each term by 5.  We get

$\displaystyle \frac{5y}{5} = \frac{-15x}{5} + \frac{20}{5}$

$\displaystyle y = -3x + 4$

We can see the slope of this line is -3.  Therefore, this line is parallel to the line 

$\displaystyle y = -3x + 1$

because it also has a slope of -3.

In order to determine if two lines have the same slope first write them according to slope-intercept form, where "m" is the slope of the line:

$\displaystyle y=mx+b$

Do that for each line:

$\displaystyle y=4x+2$

$\displaystyle y-2x=6\rightarrow y=2x+6$

The first line has a slope of 4, while the second has a slope of 2 meaning the lines are not parallel.

In order to determine if two lines have the same slope first write them according to slope-intercept form, where "m" is the slope of the line:

$\displaystyle y=mx+b$

Do that for each line:

$\displaystyle 2x+y=2\rightarrow y=-2x+2$

$\displaystyle 4x+2y=3\rightarrow 2y=-4x+3\rightarrow y=-2x+\frac{3}{2}$

Both lines have a slope of -2 which makes them parallel.

In order to determine if two lines have the same slope first write them according to slope-intercept form, where "m" is the slope of the line:

$\displaystyle y=mx+b$

Do that for each line:

$\displaystyle y-2x=3\rightarrow y=2x+3$

$\displaystyle y=2x+1$

Both lines have a slope of 2 which makes them parallel.

In order to determine if two lines have the same slope first write them according to slope-intercept form, where "m" is the slope of the line:

$\displaystyle y=mx+b$

Do that for each line:

$\displaystyle y-6x=1\rightarrow y=6x+1$

$\displaystyle 2y-12x=6\rightarrow 2y=12x+6\rightarrow y=6x+3$

Both lines have a slope of 6 which makes them parallel.

In order to determine if two lines have the same slope first write them according to slope-intercept form, where "m" is the slope of the line:

$\displaystyle y=mx+b$

Do that for each line:

$\displaystyle y=2x+3$

$\displaystyle 2y=4x+3\rightarrow y=2x+\frac{3}{2}$

Both lines have a slope of 2 which makes them parallel.

In order to determine if two lines have the same slope first write them according to slope-intercept form, where "m" is the slope of the line:

$\displaystyle y=mx+b$

Do that for each line:

$\displaystyle x-y=1\rightarrow -y=-x+1\rightarrow y=x-1$

$\displaystyle x-y=6\rightarrow -y=-x+6\rightarrow y=x-6$

Both lines have a slope of 1 which makes them parallel.

In order to determine if two lines have the same slope first write them according to slope-intercept form, where "m" is the slope of the line:

$\displaystyle y=mx+b$

Do that for each line:

$\displaystyle 6x+y=7\rightarrow y=-6x+7$

$\displaystyle 7x+2y=9\rightarrow 2y=-7x+9\rightarrow y=\frac{-7}{2}x+\frac{9}{2}$

The first line has a slope of -6, while the second has a slope of -7/2 meaning the lines are not parallel.

In order to determine if two lines have the same slope first write them according to slope-intercept form, where "m" is the slope of the line:

$\displaystyle y=mx+b$

Do that for each line:

$\displaystyle x+2y=4\rightarrow 2y=-x+4\rightarrow y=\frac{-1}{2}x+2$

$\displaystyle x-2y=4\rightarrow -2y=-x+4\rightarrow y=\frac{1}{2}x-2$

The first line has a slope of -1/2, while the second has a slope of 1/2 meaning the lines are not parallel.