Slope-Intercept and finding parallel lines I'm reviewing for my test today and I came across this question:
Which of the following lines is parallel to the line through the points $(1,3)$ and $(2,6)$?
Answers:
$a.)\:3x-y=4$
$b.)\:x-3y=4$ 
$c.)\:y=3$ 
$d.)\:x=3$ 
Now I did ${Y_2-Y_1 \over X_2-X_1}=3$. And the correct answer is $a$. My question is how do I use the number $3$ that I got to find out that it is $a$?
 A: Good job, you found that the slope of the line passing through the points $(1,3)$ and $(2,6)$ is $3$.
Now you have to remember that any line parallel to this line must have the same slope.  So we know that the slope of any line parallel to this line must have slope $3$.
If you remember your slope intercept form, when we write the equation of a line in the form $y = mx + b$, the slope of this line is $m$.
So, look at each of your answer choices, and write them each in slope intercept form.  You should get:
a) $y = 3x - 4$
b) $y = \frac{1}{3} x - \frac{4}{3}$
c) $y = 0x + 3$
d) $x = 3$
The only line in these choices with slope $3$ is choice a).  Choice b) has slope $\frac{1}{3}$, choice c) has slope $0$, and choice d) has no slope (since it is a vertical line!).
So, a) is the line parallel to our line, since it has the same slope as the line passing through $(1,3)$ and $(2,6)$.
A: Two lines are parallel (in the $xy$ plane) if their slopes are the same.
Consider to lines $y_1$ and $y_2$ such that 
$$y_1=mx+c_1,~~~~~~~~~~~~~~~~~~~~~~~ y_2=mx+c_2,$$
for some $c_1$, $c_2$ constants such that $c_1\neq c_2$, none of them $0$.
They are parallel if and only if they are never equal at any point. Suppose they are not parallel, therefore they are equal at some point $x_0$: $mx_0+c_1=mx_0+c_2 \Rightarrow c_1=c_2,\rightarrow\leftarrow$. This is a contradiction, therefore they are parallel.
What you did: $\dfrac{y_2-y_1}{x_2-x_1}=3$, means that the slope of the given line is $3$. Therefore you have to look for a line with slope $3$. This is the case of option $a)\; 3x-y=4 \Rightarrow y=3x-4$.
A: The first thing you want to do is remember are the two rules about slopes: 
1) Lines are parallel if their slopes are equal.
2) Lines are perpendicular if their slopes are the negative reciprocal.
The points you were given have a slope of, $${Y_2-Y_1\over X_2-X_1}={6-3\over2-1}=3.$$
So we want to find the other lines that also have slope 3.
Don't forget that the equation of a line is $Y=mX+b$ where $m$ is the slope and $b$ is the $Y$-intercept
So the slopes for each of the questions are given below:
$A)\ 3$
$B)\frac 13$
$C)\ 0$
$D) \ \text{undefined}$
The only one with the same slope is $A)$ so the only parallel line is $Y=3x-4$
