Algebra basic knowledge

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Introduction to Algebra


Algebra is great fun - you get to solve puzzles!

A Puzzle

What is the missing number?
 
−2=4
OK, the answer is 6, right? Because 6 − 2 = 4. Easy stuff.
Well, in Algebra we don't use blank boxes, we use a letter (usually an x or y, but any letter is fine). So we write:
x−2=4
It is really that simple. The letter (in this case an x) just means "we don't know this yet", and is often called the unknown or the variable.
And when we solve it we write:
x=6

Why Use a Letter?

 Because:
arrowit is easier to write "x" than drawing empty boxes (and easier to say "x" than "the empty box").
arrowif there are several empty boxes (several "unknowns") we can use a different letter for each one.
So x is simply better than having an empty box. We aren't trying to make words with it!
And it doesn't have to be x, it could be y or w ... or any letter or symbol we like.

How to Solve

Algebra is just like a puzzle where we start with something like "x − 2 = 4" and we want to end up with something like "x = 6".
But instead of saying "obviously x=6", use this neat step-by-step approach:
  • Work out what to remove to get "x = ..."
  • Remove it by doing the opposite (adding is the opposite of subtracting)
  • Do that to both sides
Here is an example:
We want
to remove
the "−2"
x - 2 = 4
 
To remove it, do
the opposite, in
this case add 2
x - 2 = 4 add 2 to left
 

Do it to
both sides
x - 2 = 4 add 2 to left and right
 


Which is ...
x + 0 = 6
 


Solved!
x = 6

Why did we add 2 to both sides?

To "keep the balance"...

 
balance x - 2 vs 4
In Balance
 
Add 2 to Left Side
unbalanced x - 2 + 2 vs 4
Out of Balance!
 
Add 2 to Right Side Also
balanced x - 2 + 2 vs 4 + 2
In Balance Again
Just remember this:
To keep the balance, what we do to one side of the "="
we should also do to the other side!
See this in action at the Algebra Balance Animation.

Another Puzzle

Solve this one:
x+5=12

What we want is an answer like "x = ...",
but the +5 is in the way of that!
We can cancel out the +5 with a −5 (because 5−5=0)
So, let us have a go at subtracting 5 from both sides:x + 5 −5 = 12 −5
A little arithmetic (5−5 = 0 and 12−5 = 7) becomes:x + 0 = 7
Which is just:x = 7
 Solved!
(Quick Check: 7+5=12)

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