Line L goes through point (2,13) and is perpendicular to the line y = 4x-5. Find the equation of line L.

Firstly we need to think about what facts we need to know about a line in order to write it's equation. Line equations are usually written in the form y= mx+c, which means that for any points x and y (x,y) on the line, y = m (the gradient) *x + c (intercept).  This gives you a clue of what we need to know: we need to know m (the gradient) and c (the intercept). We can then look at the information in the question to see how we can find out what the values of c and m are.

The information in the question tells us that the line (let's call it L) goes through point (2,13), and is perpendicular to y = 4x-5. Perpendicular means at right angles (90 degrees) to. Just knowing that it goes through point (2,13) doesn't give any clues about the intercept or gradient (see whiteboard). However, knowing that it is perpendicular to y = 4x-5 does (see whiteboard), because we know that the slope of line L needs to be at 90 degrees to the slope of line y=4x-5. A handy tool we can then use is that if the gradient of a line is m, the gradient of a line perpendicular to it is -1/m (the 2 gradients times together to make -1). So, in this case, the gradient of y=4x-5 is 4. (I can explain why if needed), so the gradient of a perpendicular line must be - 1/4.

Now, we know the gradient of line L. So all we need to do is find the intercept. The other piece of information we were given is that the line passes through (2,13). So at one point on the line, x = 2 and y = 13. So, if we know that for all straight lines y = mx +c, and we know m (the gradient) and we know that for one point on the line x = 2 and y=13, we can work out c (the intercept). All we have to do is substitute in m = -1/4 , x = 2, and y=13 into the general equation y = mx + c. So, 13 = (2 x -1/4) + c. Now we solve this like a normal equation for c: 13 = -0.5 +c , so c=13.5. Now, we know the gradient and the intercept: c = 13.5 and m = -1/4. So the general equation y = mx + c can be written as y = -1/4x + 13.5