Algebra is the study of unknown numbers.
With basic arithmetic, individual numbers are added, subtracted, multiplied, divided, and raised to powers or other basic functions. With algebra, however, we start looking at what happens when we compare numbers even though we do not know what specific values they have yet.
Some examples of standard equations from basic arithmetic are:
- $1+2=3$
- $2+3=5$
- $3\times3=9$
- $21+23=44$
- $7+8=15$
- $9-8=1$
- ${8\over4}=2$
- $ {2^3} = 8$
With Algebra, however, the equations begin to look different.
- $a+b=c$
- $2x+3y=z$
- $ax^2+bx+c=0$
- $(x+y)n=z$
- $a^2+b^2=c^2$
- $e=mc^2$
- $a^z+b^z=c^z | z<=2$
These equations seem slightly strange at first, but they soon become quite natural. Each letter simply stands for a unknown number Let's examine the first algebraic equation above: $a+b=c$. If we let $a=1$ and $b=2$ we find that $c=3$, so our equation reads $1+2=3$.
The power of this equation however is that we can use other values as well. If we let $a=2$ and $b=3$ then $c=5$.
We do not have to always choose $a$ and $b$ however, we could choose $a$ and $c$ and find $b$. If $a=7$ and $c=15$ we will find that $b$ must be $8$ to create a true equation.
When working with $b$ in this equation it might be easier sometimes to get it into a form where $b$ is easy to find.
$$a+b=c$$
We will change the order of addition around first
$b+a=c$
We know that $a$ does not change so we can subtract $a$ from both sides of the equation.
b+a-a=c-a
We know however that if we take
b and add
a and then subtract it again be will be back to
b. So we can remove the
+a-a from the left side of the equation.
b=c-a
Now whenever we have numbers for
c and
a,
b will be easy to find.
Much of algebra is just learning these simple techniques and applying them to more advanced problems and functions.