# Rearranging Equations

1. May 20, 2012

### hamerish

For some reason I have always had problems with rearranging equations, I have no idea how I have got so far in life without knowing how to, so iv been teaching myself.

Ita extremely simple as well, and thats why i get so worked up about them

(x-400)/(1000-400)=0.5623

I know x is 737 I just don't know how to get to it.

Any help is appreciated

Rob

2. May 20, 2012

### AmritpalS

(x-400)/(600)=0.5623
x-400=0.5623(600)
x=0.5623(600)+400
from there it's arithmetic
x=737.38~737

3. May 21, 2012

### genericusrnme

okay, I'll give you a quick lesson on the real numbers!
Here are four 'axioms' things that we take as true
1. For any non zero real number, a, there exists another real number $\frac{1}{a}$ such that $a * \frac{1}{a} = 1$
2. For any real number, b, there exists another real number -b such that b +(-b) = 0
3. Operations in the real numbers commute (this means that a * b = b * a and that a + b = b + a)
4. For any real number c, c*1 = c and c + 0 = c

We'll use these to help us solve this problem

(x-400)/(1000-400)=0.5623

This is the same as saying
$(x-400) * \frac{1}{(1000-400)} = 0.5623$

We shall first simplify the things in the brakets by doing 1000-400 = 600 to get
$(x-400) * \frac{1}{600} = 0.5623$

Next we shall invoke axiom 1 to state that there exists a number such that $\frac{1}{600} * a = 1$ and we can easily see that a must be 600 (to see this use axiom 3 and 1)
We shall then multiply both sides by 600 (we must perform the same operations to both sides to keep the equality)

$(x-400) * \frac{1}{600} * 600 = 0.5623 * 600$

Using axiom 1

$(x-400) * 1 = 0.5623 * 600$

Using axiom 4

$x-400 = 0.5623 * 600$

Using axiom 2 we find that there exists a number, a, such that -400 + a = 0, we can see that a must be 400 again, so adding 400 to both sides

$x - 400 + 400 = 0.5623 * 600 + 400$

Using axiom 2 to state 400 + (-400) = 0

$x + 0 = 0.5623 * 600 + 400$

Using axiom 4

$x = 0.5623 * 600 + 400$

Does this help?