Solving Equations

Date: 9/3/95 at 7:28:20
From: "S. Capodici"
Subject: Formulas

Can you help?
 
If n=x^2+y then how can you work out x if you know
n and y, and work out y knowing n and x.?

Is there an easy to read book on rearranging formulas?

                   Thanks, M.C.

Date: 9/4/95 at 14:31:7
From: Doctor Ethan
Subject: Re: Formulas

Hey M.C.

I guess I don't know of an easy to read book, but I can give you a few 
tips to get you started.

Two things you need to realize and remember are:

1.  If you add or multiply the same thing to both sides of an equality, it 
is still an equality.

For instance.  If 3 + b = w  then  15 + 3+ b = w + 15.

Similarly If 3+ b = w  then  17(3+b) = 17 * w.

2.  If you take the square root of both sides of an equation, then they 
are both equal in magnitude (the sign could be different).

For example:

If x^2 = 4  then  x = 2, or -2

and if 3 + r = y^2  then the square root of (3+r) equals y or -y.

From here on I will use Sqrt(3+r) to mean the square root of (3+r).

These two facts are all we need to solve the problems that you asked.

For n = x^2 + y, when we want to solve for x: First we will use rule number 1 
to get the x by itself.  We will do this by adding -y to both sides.

We then have   n - y = x^2 y-y

but y-y =0  so that reduces to   n - y = x^2.

Now we can use rule 2 and get:  Sqrt(n-y) = x or -x.

Similarly if we want to solve for y:

First we add -x^2 to both sides, and then we have

n - x^2 = x^2 +y - x^2  

which reduces to n - x^2 =y

So that is it.  I hope that this has helped some.  

-Doctor Ethan,  The Geometry Forum