There are 3 types of equation that a line can take:
This is the form you should be most familiar with, as it was the form used in GCSE maths. The 'm' is the gradient of the line, and the 'c' is the y intercept. It is the form used to find out axis intercepts.
This form is used when you are told the gradient of the line (m) and a point it passes through (x1, y1). For example, say the gradient of a line is 2 and it passes through the point (-3,7). So then you put them into the equation, getting y-7 = 2(x+3). Now, this can be changed to the y=mx+c form. Firstly, the brackets are expanded getting y-7 = 2x+6. 7 is added to both sides, getting y=2x+13. This can also be used if you are given two points; just calculate the gradient (change in y/change in x) and select one of the points to use.
This is refered to as the 'standard form' of a straight line equation. Converting from y=mx+c to it is easy. For example, with y=2x-5, just take away 2x-5 to get y-2x+5=0. Converting the other way round is a bit harder. For example, -4x+3y+7=0. First, take away (-4x+7) from both sides to get 3y=4x-7, and then divide by 3 to get y=4⁄3x-7⁄3.
When you are given a question asking to calculate the midpoint of a line, what you do is take the average of x-values, and do the same for the y-values. For example, you may be given the question 'Line AB is created by the points (-5,12) and (2,-3). Point C lies on the middle of AB. What are the coordinates of point C?' Well, the x coordinates of C are the averages of the endpoints' x-coordinates, and the same applies to the y-coordinates. Hence the coordinates of C are (-1.5, 4.5).
The best way to find out distances between any two points is to use Pythagoras; just take the horizontal and vertical distances, square them, add them together, and square root the answer
Lines are parallel if their gradients (m) are the same. Two lines are perpendicular (90 degrees) to eachother if the product of the gradients is -1, as m * -1/m = -1.