Remember that, in geometry, two lines are parallel if and only if they both lie in the same plane and never cross each other. Another way to think of that is, "they both go in the same direction". In algebra, the slope is what determines the direction a line goes so our definition of "parallel" looks like this:
Definition - Parallel Lines
Two lines are parallel if they have the same slope and their y-intercepts are different.
The part about the y-intercepts is important. If two linear equations have the same slope and the same y-intercept then they represent the same line. Instead of bein parallel and crossing at no points, they overlap each other and "cross" at every point.
There are two basic questions, you'll come across when working with parallel lines, both of which we'll go over in the examples below.
- Given the equations of two lines, decide whether or not they're parallel.
- Given the equation of a line and a point that isn't on the line, find the equation of the line through the point that's parallel to the line.
Are the lines y = 3x + 2 and x = 4y - 2 parallel?
The easiest way to do this is to start by putting both equations in the slope-intercept form, i.e. to solve both of them for y. The first equation is already in that form so we quickly see that its slope is 3 and its y-intercept is 2. The second equation still needs to be solved for y before we can get its slope and y-intercept.
x = 4y - 2
From that equation, we can see that the slope of the second line is 1/4. Because that's different from the first equation's slope, we can conclude without going any further that the lines aren't parallel.
Are the lines y = -3x - 1 and y = -3x - 7 parallel?
In this example, both lines have the same slope, -3, but there y-intercepts are different, -1 versus -3. This tells us that the lines do have to be parallel.
Are the lines 2y = 4x - 2 and y = 2x - 1 parallel?
In this example, the second equation is already in the slope-intercept form so we can see that its slope is 2 and its y-intercept is -1. To get the first equation into that form, we'll have to divide both sides of the equation by 2.
2y = 4x - 2
Looking at that equation tells us that this lines slope is also 2. Remember that, before you reach a conclusion, you also have to look at the y-intercept. Since those values are also the same for the two equations, we would conclude that the lines aren't parallel because they overlap.
Find the equation of the line that's parallel to the line y = 3x + 4 and that goes through the point (5, 7).
The first thing we need to do is look at the equation of the line that we're given and notice that its slope is 3. Since two parallel lines have the same slope that means that the slope of our line will also be 3. Now that we have the slope of the line, 3, and a point on the line, (5, 7), we can use the point-slope form of the equation of a line to get the equation of our new line.
y - 7 = 3(x - 5)