Two straight lines are parallel if they are always the same distance away from each other, no matter how long the lines are extended. In other words, they’re going the exact same direction and will never meet. Additionally, two lines are perpendicular if, when they meet, they form a right-angle. Both of these words appear all over in maths so are worth getting used to.
Before going into this topic, you should be familiar with the equation of a straight line (https://mathsmadeeasy.co.uk/gcse-maths-revision/ymxc-gcse-maths-revision-worksheets/) as well as drawing straight-line graphs (https://mathsmadeeasy.co.uk/gcse-maths-revision/drawing-straight-line-graphs-gcse-maths-revision-worksheets/).
If two lines are parallel, then they have the same gradient. This means two things, 1) you can tell if two straight lines are parallel by looking at their equations, and 2) if you are told that a line is parallel to a different line whose gradient you know, then you also know the gradient of the first line.
Example: Is the line y=3x-4 parallel to the line 3y-9x=21?
So, we need to know the gradient of both lines. The first line equation is given to us in the desired form of y=mx+c, so we know its gradient (the coefficient of x) is 3.
The second line equation requires some rearranging before we can know its gradient. Firstly, add 9x to both sides of the equation 3y-9x=21 to get
Then, if we divide both sides by 3 we get
This is now in the right form, and we can see that its gradient is 3 so must be parallel to the 1st line.