In order to solve the equation for this line, you need to things: the slope, and at least one point. You are already given a point for the line, so you just need to figure out the slope. The other piece of information you have is a line parallel to the line that you're looking for; since parallel lines have the same slope, you just need to figure out the slope of the parallel line you've already been given. 

To figure out the slope, change the equation into point-slope form (y = mx+b) so the slope m is easy to find.  To do that, you need to isolate y on one side of the equation.

By the calculations above, you'll find that the slope of the parallel line is -1/2.  

Now, use this slope of -1/2 and the point 4,1 to find the equation. First, plug them both into the point-slope form, then solve for the slope-intercept form.

We start by rearranging the equation into the form y = mx + b (where m is the slope and b is the y intercept); y = 4x – 22
Now we know the slope is 4 and so the equation we are looking for must have the m = 4 because the lines are parallel. We are also told that the equation must pass through the origin; this means that b = 0.

In 4x – y = 0 we can rearrange to get y = 4x. This fulfills both requirements.

The given equation is in standard form and needs to be converted to slope-intercept form which gives y = –2/5x + 6/5. The parallel line will have a slope of –2/5 (the same slope as the old line). The slope and the given point are substituted back into the slope-intercept form to yield y = –2/5x +5.

The slope of the given line is  and a parallel line would have the same slope, so we need to find a line through  with a slope of 2 by using the slope-intercept form of the equation for a line.  The resulting line is  which needs to be converted to the standard form to get .

Since they are parallel, the line will have the same slope as .

 Thus, it will take the form .

We then use the point (3,2) to solve for :

so the equation of the line is

In this equation,  is equal to the slope of the line.  You have been given a line parallel to the line containing variable , and since parallel lines have the same slope, all you need to do is figure out the slope of  in order to figure out what  is.  To figure out the slope of , you need to convert the equation into  form, which means isolating  on one side of the equation.

Given this solution, the slope of the parallel lines, and thus , is equal to 2.

This prompt asks you to find a line that is parallel to the one given. Parallel lines have no points of intersection (as opposed to intersecting lines which have 1 point of intersection). Parallel lines have the same slope, but different y-intercepts. If they had the same y-intercept, then they would actually be the same line. 

We can find the slope of the first line by converting it to slope-intercept form. 

First, subtract 3x from both sides.

Then, divide both sides by -2.

The slope of the line is 3/2. The only answer that has that same slope, but a different y-intercept is  .

The slope of the line is  and the y-intercept is .

The equation of the line is . 

This is the parent graph of . Since the graph in question is negative, then we flip the quadrants in which it will approach infinity. So the graph of  will start in quadrant 2 and end in 4. 

Plug in  for  to find a point on the line:

Thus,  is a point on the line.

Plug in   for  to find a second point on the line:

 is another point on the line.

Now we know that the line passes through the points  and .  

A quick sketch of the two points reveals that the line passes through all but the third quadrant.