In beginning Algebra, we let the variables stand for some real number. Thus the properties of real numbers can be used when dealing with the variables.
Some important properties:
commutative property: `2*3=3*2` so `(x+2)x=x(x+2)`
associative property: (2+3)+5=2+(3+5) so (x+3)+5=x+(3+5)
identity: 3+0=0+3=3 x+0=0+x=x This idea is used when studying quadratics and learning to complete the square.
`3*1=1*3=3` and `x*1=1*x=x` I learned to call certain things FFOO's (pronounced fahfooh) or a fancy form of 1. You use this idea wn rationalizing, finding common denominators, and many other places.
inverse: 3+(-3)=0 and x+(-x)=0
`3*1/3=1` and `x*1/x=1,x!=0` We use these all of the time to transform equations. We add/subtract the same thing to both sides of an equation (or multiply/divide). We are then left with an addition of zero or multiplication by 1. (This is not frequently emphasized.)
e.g. x+5=7 Subtract 5 (or add -5) to both sides to get x+5-5=7-5 or x+0=2 (using the inverse property) or x=2 (using the identity property.)
distributive property: 2(3+5)=2(3)+2(5) so 2(x+3)=2x+6 used on virtually every problem.
These properties form the logical basis that allows you to perform operations on the numbers and variables you encounter.