Algebra - Introducing Simple Equations
Many people think that equations and algebra are beyond them - the thought of having to work with equations fills them with fear. However, there is no need to be afraid of equations.
The good news is that equations are actually relatively simple concepts, and with a bit of practice and the application of some simple rules, you can learn to manipulate and solve them.
This page is designed to introduce you to the basics of algebra, hopefully making you feel more comfortable solving simple equations.
What is an Equation?
An equation is two expressions on either side of a sign that indicates their relationship.
That relationship may be equals (=), less than (<) or greater than (>), or some combination (for example, less than or equals to <=).
Simple equations therefore include 2 + 2 = 4 and 5 + 3 > 3 + 4.
However, when most people talk about equations, they mean algebraic equations.
These are equations that involve letters as well as numbers. Letters are used to replace some of the numbers where a numerical expression would be too complicated, or where you want to generalise rather than use specific numbers.
Algebraic equations are solved by working out what numbers the letters represent.
We can turn the two simple equations above into algebraic equations by substituting x for one of the numbers:
2 + 2 = x
We know that 2 + 2 = 4, which means that x must equal 4. The equation answer is therefore x = 4.
5 + 3 > 3 + x
We know that 5 + 3 = 8. The equation tells us that 8 is greater than (>) 3 + x.
Take 3 away from 8, to get 5.
We can see that x must be less than 5 or x is 4 or less. x < 5 or x <= 4
We cannot say more precisely what x is with the information that we are given.
There is no magic about using the letter x. You can use any letter you like, although x and y are commonly used to represent the unknown elements of equations.
Variables and constants
A letter used to substitute for a number in algebra is called a variable, because it stands for different numbers each time you use it.
This is different from a particular letter which is always used to substitute for the same number, such as π (pi) which is always 3.142. Such a letter is called a constant.
In an algebraic equation, any numbers are also constants, because they always stay the same.
If you are required to do a sum involving a constant, you will always be told its value.
Terms of an Equation
A term is a part of the equation that is separated from other parts by an addition or subtraction sign.
Terms may be just numbers, or they may be just letters, or they may be a combination of letters and numbers, such as 2x, 3xy or 4x2.
In a term involving letters and numbers, the number is known as the coefficient, and the letter as the variable.
Terms that have exactly the same variable are said to be like terms, and you can add, subtract, multiply or divide them as if they were simple numbers. For example:
The equation 2x + 3x is equal to 5x, simply 2 lots of x plus 3 lots of x to make 5 lots of x (5x).
5xy –xy = 4xy
5y × 3y = 15y.
You cannot add or subtract unlike terms. However, you can multiply them by combining variables and multiplying the coefficients together.
So, for example, 3y × 2x = 6xy (because 6xy simply means 6 times x times y).
You can divide unlike terms by turning them into fractions and cancelling them down. Start with the numbers, then the letters.
So, for example, 6xy ÷ 3x =
|The 1 can be
by 1 is itself
Rearranging and Solving Equations
In many cases to solve an equation you will probably need to rearrange it, that is, to move the terms about inside it so that you end up with only terms involving x on one side of the sign and all the numbers on the other.
This process is sometimes called isolating x.
You can rearrange equations through a set of simple rules:
- Whatever you do to one side of the equation, you must do the same to the other. That way you preserve the relationship between them. It doesn’t matter what you do, whether it’s take away 2, add 57, or multiply by 150. As long as you do it to both sides, the equation remains correct.
- Our page on Addition explains that it doesn’t matter what order you add in, the answer is still the same. This means that you can move the sum about to put the like terms together and make it easier to add up. This applies to Subtraction too as long as you remember from our page on Positive and Negative Numbers that subtracting is the same as adding a negative number. So, for example, 10 − 3 = 10 + (-3).
- Equations work according to BODMAS too, so remember to do the sum in the right order.
- Always get your equation into the simplest possible form: multiply out brackets, divide down, cancel out fractions, and add/subtract all the like terms.
Try to solve these equations, click on the boxes to reveal the workings and answers.
- As with any sum, do the multiplication first. 5 × 4 = 20
- So x + 3 = 20
- The next step is to take three away from both sides
- x + 3 - 3 = 20 - 3
- 20 - 3 = 17.
x therefore equals 17
- Do the sum on the right hand side first, because it doesn’t involve any letters. There are no brackets, so it’s multiplication first, then addition.
- 6 × 5 = 30, and 30 + 3 = 33.
- The sum on the left is an addition one, so you can move the terms about, until you have all the numbers together:
5 + x + 21 = x + 5 + 21
5 + 21 = 26.
- So now you have 26 + x = 33
- Now you can take 26 away from both sides
- 26 + x - 26 = x = 33 - 26
- 33 - 26 = 7.
Therefore x = 7
- Rearrange to get all the numbers on one side, by taking five away from each side.
- Now you have
x2 = 13 - 4 - 5 = 4
- Now you need to take the square root of both sides, because you want to know what x is, not x2.
- You know that 2 × 2 = 4, which means that the square root of 4 = 2
x = 2
Equations and Graphs
Any equation in which there is a relationship between just two variables, x and y, can be drawn as a line graph where x goes along the horizontal axis (sometimes called the x axis) and y on the vertical axis, (sometimes called the y axis).
You can work out the points by solving the equation for particular values of x.
y = 2x + 3
|sum||2(0) + 3||2(1) + 3||2(2) + 3||2(3) + 3||2(4) + 3||2(5) + 3||2(6) + 3|
The advantage of drawing a graph of an equation is that you can then use it to work out the value of y for any given value of x, or indeed x for any given value of y, by looking at the graph.
In this example what is the value of x when y = 10?
Move up the y axis until you reach 10, then move out at that level horizontally until you reach the graph. At that point, move downwards until you reach the x-axis. From the red lines in the graph, you can see that when y = 10, x = 3.5.
y = x2 + x + 4
When x = 0, y = 0 + 0 + 4, when x = 1, y = 1 + 1 + 4 = 6 and so on...
Another advantage of plotting the answers to an equation on a graph is that you can extrapolate graphs (extend them beyond the numbers that you have actually worked out) to work out bigger values of x or y.
However, care is needed when extrapolating a graph that isn't a straight line.
This page has explained how to solve simple equations, and the relationship between equations and graphs, giving you an alternative way to solve equations.
You are now ready to move onto more complex equations, including simultaneous equations and quadratic equations.