How do I find the equation of a line that is perpendicular to another line?

Okay so let's say that the equation of the line you are given is y=2x+3 and you are told this new line must pass through the coordinates (2,4). The first thing we need to do is check we understand every bit of the question. If the new line that we want to find is perpendicular to the one we are given, what does this tell us about its direction or how the lines meet? It tells us that they meet at a right angle. This means that the gradients are complete opposites. In maths terms our new gradient is the negative reciprocal of the gradient of the first line. So the gradient of the line they have given us is 2, because thats the number that we see infront of the x. To get our new gradient, the negative reciprocal we have to "flip it" and times by -1, so 2 becomes -1/2 (this is our gradient of the new line).

To use the next bit of information in the Question, the coordinates (2,4), we can use a very useful formula: y - y₁= m (x - x₁) this formula, once we have put all the useful information in, will "magically" produce the equation for our new line! In this formula m= the new gradient= -1/2,  y₁= the y coordinate= 4, and x₁= the x coordinate= 2. Put this all in the formula and we have: y - 4= -1/2 (x - 2). We then expand the brackets: y - 4= -x/2 + 1. We then make sure that it just says y= on one side and we have the final answer! : y = -x/2 + 5