\(\displaystyle y=2x+4\)

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.

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

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

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

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

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