Algebra

Algebra is a huge part of mathematics.  A lot of students hear about it a long time before they actually have to do any algebra.  It’s not very hard to learn how to do, especially if you get to understand the basics early on.  And once you know how to ‘do algebra’, you will be able to solve a large variety of challenging mathematics problems easily. 

The first and biggest thing to understand is what a variable is in algebra.  Say I have the following equation:

                                                          

This is a really simple equation – everyone knows that “two plus seven equals nine”.  This equation has three numbers in it – a ‘2’, a ‘7’ and a ‘9’.  It also has two operations – a ‘+’ sign and an ‘=’ sign. 

Now, instead of one of these numbers, let’s write a letter in its place:

                                                        

Notice how now, instead of the number ‘2’, there is a letter ‘x’.  If you read this new equation out aloud, you should say something like: “x plus seven equals nine”.  Notice that instead of saying ‘two’ we now say ‘x’.  ‘x’ is what we call a variable, or pronumeral.  Variables are used to represent numbers. 

So in this case, we know that the ‘x’ represents the number ‘2’, since we replaced the ‘2’ with the ‘x’.  So if we wanted to write the equation again, but this time writing it with the number that ‘x’ is representing, we’d write a ‘2’ where the ‘x’ is now:

                                                        

Notice how we’ve ended up with our original, all number equation.

In this last example we used a letter (‘x’) as a variable.  Variables can also be symbols.  People sometimes use Greek Symbols as variables, such as θ, called ‘theta’.  If we used θ instead of ‘x’ in the last equation, we’d write:

                                                        

This time the ‘2’ would be represented by θ.

Now, what happens if we start with an equation which already has a variable in it:

                                                          

This equation has the variable ‘x’ in it.  This time though, we don’t already know what number the ‘x’ is representing.  If we want to find out what ‘x’ represents, we need to “solve the equation for x”.  This means – work out what value the ‘x’ is representing.

To solve the equation for x, we need to work out what number we can replace the ‘x’ with to make the equation true.  So, we might as well try a number – let’s try ‘1’:

                                                         

So instead of writing ‘x’, I wrote ‘1’ down, and then calculated what 5 – 1 equals – it equals 4.  We are trying to get 2 as our answer, not 4, so using 1 does not make the equation true.

Let’s try replacing ‘x’ with ‘2’:

                                                         

So using ‘2’ instead of ‘x’ doesn’t make the equation true either.  When we put ‘2’ instead of ‘x’, we get an answer of 3, and we want an answer of 2.  But we’re on the right track, cause 3 is only just bigger than 2.  Let’s try replacing ‘x’ with ‘3’:

                                                         

Bingo!  When we use ‘3’ to replace ‘x’, we get an answer of ‘2’, which is what we want.  We can say something like: “When x equals three, the equation is true.”  If you were writing an answer to the question, “Solve the equation for x,” you’d write:

                                                             

Coefficients of variables

A coefficient is the number in front of a variable.  Look at the following expression:

                                                               

The number in front of the ‘x’ is its coefficient: 5. What about:

                                                         

The coefficient of ‘x’ is ‘–4’ and the coefficient of y is ‘3’. 

When I write something like , all I’m saying is that I have “five lots of x”.  So if I have an equation like:

                                                           

I can read this equation as “Three lots of x plus four lots of x”.  What happens when I have 3 of something, and I add another 4 of the same thing?  Simple – I end up with 7 of that thing.  So in this case, I can rewrite this expression:

                                                         

What about if I have something like:

                                                           

Can I add these two together?  The answer is NO.  I can only add together variables that are the same.  ‘x’ and ‘y’ are different variables, so I can’t add them together.

Multiplying and dividing variables

You can do other things with variables apart from add or subtract them.  For instance, variables can be multiplied and divided – say I wanted to multiply ‘x’ by ‘y’.  I’d write this as:

                                                               

When you’re multiplying variables together, often the multiplication symbol isn’t written.  The last expression is exactly the same as:

                                                             

except it doesn’t have the multiplication symbol.

You can also divide variables by other variables:

                                           

are all ways of writing “x divided by y”. 

Expressions, equations and terms

Some students (such as me) have problems remembering the difference between an expression and an equation.  To understand the difference, you also need to know what a term is.  A term is part of an equation which has one or more variables or pronumerals in it, but no ‘+’ signs or  ‘–’ signs in it.  For instance:

·          is a term because it has two pronumerals in it, a and b. 

·          is also a term because it has the two pronumerals x and y in it, but no ‘+’ or ‘–’ signs in it.

·          is not a term because it has a ‘+’ sign in it.  In fact, it is made up of two terms, ‘4x’ and ‘3y’.

What about something like ‘5’ – is this a term?  Numbers by themselves are often called numeric terms – so for something like: , there are two terms – one algebraic or normal term (3x), and one numeric term (5).

So now that we know what a term is, we can understand what expressions and equations are.

An expression is a group of terms connected by ‘+’ or ‘–’ signs.  There can be as many terms as you want, as long as there is at least 1.  For instance,

                                                     

is an expression, because it contains three terms, which are joined together by a ‘+’ and a ‘–’ sign.

Expressions do not have ‘=’ signs in them.  Equations do however.  Now that we understand what expressions are, we can use them to help understand what an equation is.

An equation is made up of two expressions with an ‘=’ sign between them.  One expression is on the left of the ‘=’ sign, and the other expression is on the right of the ‘=’ sign.  Here’s one example of an equation:

                                                     

Notice that the equation is made up of two expressions – ‘3x + 4’ and ‘5x – 2’.  Also see how they are joined together by an ‘=’ sign.  Equations tell the reader that all the stuff on the left of the ‘=’ sign is the same as all the stuff on the right of the ‘=’ sign.

Like terms

Like terms contain the same variables and numbers, raised to the same powers.  For instance,

                                                 is the same as

because both terms contain the same number and the same variables.  What about:

                                                    and ???

These are not like terms.  They both have the same number – ‘3’.  They also have the same variables – ‘x’ and ‘y’.  However, the variables are raised to different powers in each term.  The first term has ‘x’ raised to the 2^nd power, but not in the second term.  Also, the second term has ‘y’ raised to the 2^nd power, but this doesn’t happen in the first term.

Binomial and trinomial expressions

There are two common types of expressions, called binomial and trinomial expressions.  You can guess what these expressions mean by looking at their names.  The ‘bi’ part of binomial means ‘two’ – binomial expressions have two terms.  Now look at the ‘tri’ – tri means ‘three’ – trinomial expressions have three terms.  So here’s an example of each type of expression:

                                           is a binomial expression.

                                    is a trinomial expression.