$\displaystyle y=2x+4$

A parallel line will have the same $\displaystyle m$ value, in this case $\displaystyle 2$, as the orignal line but will intercept the $\displaystyle y-axis$ at a different location. 

Parallel means the same slope:

$\displaystyle 3y=x-2$

$\displaystyle y=mx+b$

Solve for $\displaystyle m$:

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

$\displaystyle m=\frac{1}{3}$

Find the linear equation where

$\displaystyle m=\frac{1}{3}$.

$\displaystyle y=\frac{x}{3}+72$

The line parallel to $\displaystyle y=4x+2$ will have the same slope. 

The equation for our parallel line will be:  $\displaystyle \ y=4x+b$. 

Using the point (1,1) we can solve for the y-intercept:

$\displaystyle 1=4(1)+b$

$\displaystyle \ 1=4+b$

$\displaystyle 1-4=b$

$\displaystyle -3=b$

Parallel lines have identical slopes. If you convert the given equation to the form $\displaystyle y=mx+b$, it becomes 

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

The slope of this equation is $\displaystyle -\frac{1}{2}$, so its parallel line must also have a slope of $\displaystyle -\frac{1}{2}$. The only other line with a slope of $\displaystyle -\frac{1}{2}$ is 

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

Parallel lines have identical slopes. To determine the slope of the given line, convert it to $\displaystyle y=mx+b$ form:

2y = 3x + 8

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

This line has a slope of $\displaystyle \frac{3}{2}$.

The only answer choice with a slope of $\displaystyle \frac{3}{2}$ is $\displaystyle y=\frac{3}{2}x-6$.

$\displaystyle 2y=\frac{3}{2}x+3$ is the correct answer because when each term is divided by 2 in order to see the equation in terms of y, the slope of the equation is $\displaystyle \frac{3}{4}$, which is the same as the slope in the given equation. Parallel lines have the same slope. 

The equation of a line can be written using the expression $\displaystyle y=mx+b$ where $\displaystyle m$ is the slope and $\displaystyle b$ is the y-intercept. When lines are parallel to each other, it means that they have the same slope, so $\displaystyle m=-\frac{1}{2}$. The y-intercept is given in the problem as $\displaystyle -5$. This means that the equation would be $\displaystyle y=-\frac{1}{2}x-5$.

In order to approach this problem, we need to be familiar with the slope-intercept equation of a line, $\displaystyle \small y=mx+b$ where m is the slope and b is the y-intercept. The line that our line is supposed to be parallel to is $\displaystyle \small y=\frac{1}{4}x-5$. Lines that are parallel have the same slope, m, so the slope of our new line is $\displaystyle \small \frac{1}{4}$. Since we don't know the y-intercept yet, for now we'll write our equation as just:

$\displaystyle \small y=\frac{1}{4}x+b$. We can solve for b using the point we know the line passes though, $\displaystyle \small (4, -2)$. We can plug in 4 for x and -2 for y to solve for b:

$\displaystyle \small -2 = \frac{1}{4}*4 + b$ first we'll multiply $\displaystyle \small \frac{1}{4}*4$ to get 1:

$\displaystyle \small \small -2 = 1 + b$ now we can subtract 1 from both sides to solve for b:

$\displaystyle \small -3 = b$

Now we can just go back to our equation and sub in -3 for b:

$\displaystyle \small y = \frac{1}{4}x -3$

Parallel lines have the same slope. So our line should have a slope of 2x. Next we use the point slope formula to find the equation of the line that passes through $\displaystyle (5,5)$ and is parallel to $\displaystyle y=2x+5$.

Point slope formula:

$\displaystyle y-y_{1}=m(x-x_{1})$

$\displaystyle y-5=2(x-5)$

$\displaystyle y-5=2x-10$

$\displaystyle y=2x-5$ is the slope of the line parallel to $\displaystyle y=2x+5$ which passes through $\displaystyle (5,5)$.

Parallel lines have the same slope, so the slope of the new line will also have a slope $\displaystyle m=-\frac{3}{2}$

Use point-slope form to find the equation of the new line.

$\displaystyle y-y_{1}=m(x-x_{1})$

Plug in known values and solve.

$\displaystyle y-(-5)=-\frac{3}{2}(x-2)$

$\displaystyle y+5=-\frac{3}{2}x-\frac{-(3)}2{}2$

$\displaystyle y+5=-\frac{3}{2}x+3$

$\displaystyle y=-\frac{3}{2}x+3-5$

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