How do you solve simultaneous equations and why do you do it?

Solving simultaneous equations allows you to identify at which point on a graph two lines intersect. For example, take the straight line -x + y = 2 and parabolic curve y= x²  Firstly, let's re-arrange the straight line equation into a more familiar form of y = mx + c. This is done by keeping the y on one side and moving the x over, remembering to change the sign as you subtract the x across, giving y = x + 2. By doing this we can now use the elimination technique, which allows us to remove y from the equation, leaving us with just one unknown, instead of the original two. This is because y is the same in both equations, so x+2 = x² Now we can tidy this equation up, by collecting like terms and equating it to zero. Ideally, we should keep x² positive for simplicity, so that means shifting everything else to the right-hand side, again remembering to change signs as you cross over. So this equation can be simplified to x² - x - 2 =0. Now, we see if we can factorise the quadratic into the form (x+a)(x+b), where a and b could be either positive or negative.The key here is to remember that ab must equal -2 and ax + bx must equal -1. So numbers which multiply together to make -2 are: -1 and 2; -2 and 1. Plugging these into ax + bx = -1, then it's clear that a=-2 and b=1. So the factorised equation is (x-2)(x+1)=0. For this expression to be true, then one of the brackets must equal zero. So either x=2 or x=-1. Finally, you can plug these x values back into either equation (they should give the same answer) to give the corresponding y value, and thus give the points of intersection of the two lines. For example: y=x² So when x=2 y=4 and x=-1 y=1. So the points of intersection of these two lines are (2,4) and (-1,1). Although we used the elimination method here, you could also use the substitution method which will give an identical answer.