Introduction to AlgebraAlgebra is great fun - you get to solve puzzles!

A PuzzleWhat is the missing number?

  - 2 = 4OK, 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 = 4It 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:

it is easier to write "x" than drawing empty boxes (and easier to say "x" than "the empty box").

if 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 SolveAlgebra 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:

Here is an example:

We want to remove the "-2"

To remove it, do the opposite,

in this case add 2:Do it to both sides: Which is ... Solved!

Why did we add 2 to both sides?

To "keep the balance"...

In Balance

Add 2 to Left Side

Out of Balance!

Add 2 to Right Side Also

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!

Another PuzzleSolve this one:

x + 5 = 12

Start with: x + 5 = 12

What we are aiming for is an answer like "x = ...", and the plus 5 is in the way of that!

We can cancel out the plus 5 by doing a subtract 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)

Algebra - Basic DefinitionsIt may help you to read  Introduction to Algebra  first

What is an EquationAn equation says that two things are equal. It will have an equals sign "=" like this:

x + 2 = 6

That equation says: what is on the left (x + 2) is equal to what is on the right (6)

So an equation is like a statement "this equals that"

Parts of an EquationSo people can talk about equations, there are names for different parts (better than saying "that thingy there"!)

Here we have an equation that says 4x − 7 equals 5, and all its parts:

A Variable is a symbol for a number we don't know yet. It is usually a letter like x or y.

A number on its own is called a Constant.

A Coefficient is a number used to multiply a variable (4x means 4 times x, so 4 is a coefficient)

Sometimes a letter stands in for the number:

Example: ax2 + bx + c

x is a variable a and b are coefficients c is a constant

An Operator is a symbol (such as +, ×, etc) that shows an operation (ie we want to do something with the values).

 

A Term is either a single number or a variable, or numbers and variables multiplied together.

An Expression is a group of terms (the terms are separated by + or − signs)

So, now we can say things like "that expression has only two terms", or "the second term is a constant", or even "are you sure the coefficient is really 4?"

Exponents

The  exponent  (such as the 2 in x2) says how many times to use the value in a multiplication.

Examples:

82 = 8 × 8 = 64

y3 = y × y × y

y2z = y × y × z

Exponents make it easier to write and use many multiplications

Example: y4z2 is easier than y × y × y × y × z × z, or even yyyyzz

PolynomialExample of a Polynomial: 3x2 + x - 2

A  polynomial  can have constants, variables and the exponents 0,1,2,3,...

But it never has division by a variable.

Monomial, Binomial, TrinomialThere are special names for polynomials with 1, 2 or 3 terms:

Like TermsLike Terms  are terms whose variables (and their  exponents  such as the 2 in x2) are the same.

In other words, terms that are "like" each other. (Note: the coefficients can be different)

Example:

(1/3)xy2 -2xy2 6xy2

Are all like terms because the variables are all xy2

A PuzzleWhat is the missing number?

  × 4 = 8The answer is 2, right? Because 2 × 4 = 8.

Well, in Algebra we don't use blank boxes, we use a letter. So we might write:

x × 4 = 8 

But the "x" looks like the "×"  ... that can be very confusing ... so in Algebra we don't use the multiply symbol (×) between numbers and letters:

We put the number next to the letter to mean multiply:

4x = 8In English we say "four x equals eight", meaning that 4 x's make 8.

 

And the answer is written:

x = 2

How to SolveInstead of saying "obviously x=2", use this neat step-by-step approach:

Work out what to remove to get "x = ..." Remove it by doing the opposite Do that to both sides

And what is the opposite of multiplying? Dividing!

Have a look at this example:

We want to remove the "4"

To remove it, do the opposite, in 

this case divide by 4:

Do it to both sides: Which is ... Solved!