HOW TO MANIPULATE EQUATIONS --------Warren Smith--------- Suppose you have an equation A=B, where A and B both might be some large expressions. LEGAL THINGS TO DO TO THE EQUATION: ----------------------------------- 0. You can SIMPLIFY either side. 1. You can add something (call it C) to both sides: A+C = B+C. (If C is negated this is subtracting C from both sides, so that is really same thing.) 2. You can multiply both sides by something: A C = B C. [warning. If C=0 this is legal but just leads to 0=0 which is not a useful equation, and note in this case we get equality even if it was not originally there.] If C is reciprocated this is dividing C from both sides, so that is really same thing - except now C is not allowed to be 0 since we are not allowed to divide by 0. 3. You can do the SAME thing (ANYthing represented by a function named F) to both sides: F(A) = F(B). [Warning. If F does not have an inverse because it is a many-to-one function, i.e. its inverse has multiple "branches", then this, while legal and causing any equality to stay an equality, can cause non-equalities to become equalities. For example if you square both sides of the non-equality -2 = 2, you get 4=4.] 4. You can do a SUBSTITUTION, also called a CHANGE OF VARIABLES: every time there is a letter (say x) in your equation, you can replace it by any fixed expression (say y). Also, you can replace some of them by y and some by z, if you happen to know from some other equation that y=z. Example. In the equation 2 x + 6 x + 5 = 0 we substitute x = y-3. (Or we "change variables from x to y where y = 3+x." Same thing.) We get 2 (y-3) + 6 (y-3) + 5 = 0. Now if we expand this out it is the same as 2 y - 4 = 0 which is an easy equation to solve for y (y = +2 or y = -2) and then x = y-3 must be -5 or -1. WHY YOU SHOULD DO THESE THINGS, AND WHICH TO DO: ------------------------------------------------ You should choose legal moves to make which cause your equation to get closer to some desired form. If your goal is to solve for x, then the desired form has x on one side of the equation and all the other stuff on the other side. 7 tan(x) + 3 = 59 subtract 3: 7 tan(x) = 56 times 1/7: tan(x) = 8 take arctan: x = arctan(8). <-- actually really x + n pi = arctan(8), n any integer, due to many-branchedness and pi-periodicity of tan(x). If we knew x was in (0, pi), though, what I originally said was ok. You can change the names of variables, or subexpressions, in your formula to make your life easier. For example if there were some large subexpression, like tan(x) + 5 x + 99/sin(x+7) occurring a lot inside your equation, you might just call it S, then you have a much easier equation to write down, and manipulate THAT. At the end you can restore it. If you have several equations, and your goal is to "express x in terms of a and b" that means you want an equation with x on one side and some expression involving a and b only, on the other side. If the original equation involves extraneous stuff such as c and d, that means we have to express c and d in terms of a and b only, as a subgoal, in order to accomplish the goal. Example: x a + b + c = 6 and (a+5) c = b + a. So we solve for c first (in the righthand equation): c = (b+a)/(a+5) then substitute this into the first equation to get x a + b + (b+a)/(a+5) = 6 then we have got rid of ("eliminated") c, and can now solve for x x a = 6 - b - (b+a)/(a+5) so x = [ 6 - b - (b+a)/(a+5) ] / a. done. SANITY CHECKING --------------- Especially for beginners, manipulating equations can be error prone. Often an easy way to reduce your number of errors dramatically, is to do "sanity checks". That means: if you have some (alleged) equation, you find a simple-enough special case of it that you can test it numerically. Example: Suppose you solve some equation F(x)=G(x) for x. The solving process was long and error-prone. Now you have your answer (you hope). Find the numerical value of the answer, say 2.13. Now test whether F(2.13)=G(2.13) is really true. But sometimes you cannot "find the numerical value of the answer" because the "answer" is really itself a formula depending on some free parameters. For example say the goal of some long manipulation was to express something (call it x) in terms of some other stuff (call it a,b). OK, when you get done you have x = F(a,b) (with hopefully correct expression for F). Well, you can do a sanity check by picking some random (or especially simple and clearly legal) numerical values for a,b, then test if the original equations worked for those a,b, values using the x-value x=F(a,b). For example if I tell you "simplify this expression F(a,b,c,d)" and you do a very long chain of simplifications to get an "answer" S(a,b,c,d), you know that F(1.5, 3.3, 1.1, 2.9) = S(1.5, 3.3, 1.1, 2.9) i.e. the two expressions must be the same at any particular set of (the same) random parameter values. Sanity check: Just run them both thru your calculator to see if really do get the same values. (It helps to use simpler numbers.) This kind of thing is also very useful when you almost remember some formula, but are unsure of some part of it. You can often easily rule out the wrong choices by means of sanity checks. For example if you thought the area of a rectangle was Base+Height (WRONG, it is base TIMES height) you could do, as a sanity check, a rectangle with base=1 and height=1, which you can just see has area=1. I mean, you have to know the area of SOMETHING, right? Use the example you know as the sanity check. 1+1 does not pass this sanity check. 1 times 1 does. Sanity checks do not ALWAYS spot errors, for example the area of a 2 times 2 rectangle actually IS 2+2, by (un)lucky coincidence. But they usually detect errors. ELIMINATION ----------- If you have N equations in N unknowns then you can (1) solve one equation for one unknown (call it x) in terms of the other N-1 (2) SUBSTITUTE that into the other N-1 equations, thus "eliminating" x , we now have N-1 equations in N-1 unknowns (no x's in these N-1 equations anymore.) Keep eliminating, N --> N-1 --> ... --> 1 and we end up with 1 equation in 1 unknown. Solve it, then go backwards to get all the other unknowns. ILLEGAL WRONG THINGS NOT TO DO: ------------------------------- ln(1) ----- = 0. It does NOT equal 1/2 by "cancelling the ln's"!!! ln(2) Remember, only cancel things you multiply AND divide by, since the net effect is same as doing nothing. ln(x) is not a multiplication, it is taking a function value, the natural log, of x, written ln(x) and available on many calculators via a "ln" button.