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.
