Expressions

Definitions

Learn these definitions first.

A term is a part of an expression, which can be a variable or a constant.

Constant – A constant is a value in maths that will never change. e.g., 1, 2, 3, π

Variable – A variable is a value in maths that can change. Its value varies. x, y, z

Application

Think of the number of students in each form group. Will it always be the same? No, some classes will have more people than others. So, the number of students in each form group will change – it is not a fixed number. For example, some classes will have 30, and some might have 28. So, it is a variable number. We can represent it with a letter like x.

Now, think of the number of days in a week. There are 7 days in a week – this will never change. In other words, the number of days in a week is constant.

Like Terms

In algebra, when we have two or more like terms, we can add, subtract, multiply, or divide them like any other number. Like terms are terms that have the same variables.

For example, you can add 7x and 4x to get 11x. But 7x + 3y cannot be simplified further because the variables are different.

Note that 1x is exactly the same as x. Think about it: x can be anything. If you have only one lot of x, it is the same as 1x, so you can simply write x.

Why This Works

Think of it like this: As you know, x can change. "x" is just a label we give to a quantity that can change. Let's say we wanted to label each fruit in a bowl. Instead of calling these fruits x, we could call them by their actual names.

So, we want to see how many apples and oranges there are.

We see 2 apples + 2 apples + 3 oranges.

So, we have 4 apples and 3 oranges. We can add the apples together, but because we know apples are not oranges, we can't add them together.

Now, replace apples with x and oranges with y. We have 2x + 2x + 3y, so we can add the like terms of x to get 4x + 3y. This is 4 apples and 3 oranges.

Different Like Terms rules

So the rule for doing anything to like terms is that the variable letters or constants after the number are the same.

2x + 2x is 4x

2x + 2y can't be simplified further because the variable letter after each number isn't the same (x ≠ y)

It doesn't matter how many letters are after the number, this rule always works:

2abcd + 4abcd simplifies to 6abcd

Also, here are some general rules:

2x + x = 3x

2x - x = x

When you do something like 3x-5x you get a negative result, -2x. This is the same as normal subtraction, just don't forget the variables.

If the numbers divide two like terms together, the variable disappears and you divide the constants.

4x ÷ 2x = 2

Just remember to remove variables and divide constants.

4x × 2x = 8x²

Multiplying two like terms: You multiply the constants and get ², which means squared (more on this later).

Now Try This:

Complicated Like Terms

Step by step plan

1. Circle and Count each term
2. Separate each like term from the other
3. Process each like term by adding/subtracting/multiplying/dividing, remember BIDMAS.
4. Repeat - until you've covered each term

Example

Now Try This:

Useful Practice questions

Here is a very useful set of questions from corbett maths. Answer them all and mark to see how well you know this topic.

Summary

• Understanding Terms, Constants, and Variables:
  • Terms: A part of an expression, which can be a variable or constant.
  • Constants: Fixed values that do not change, e.g., 1, 2, 3, π.
  • Variables: Values that can change, represented by letters like x, y, z.
• Like Terms:
  • Like terms can be combined by adding or subtracting them if they have the same variable.
  • Example: 7x + 4x = 11x, but 7x + 3y cannot be simplified further because the variables differ.
• Rules for Simplifying Like Terms:
  • When like terms have the same variable, they can be added, subtracted, multiplied, or divided.
  • Multiplying like terms, e.g., 4x × 2x = 8x².
  • Dividing like terms, e.g., 4x ÷ 2x = 2.
• Complicated Like Terms:
  • Follow a step-by-step plan: Circle and count terms, separate like terms, apply operations, and repeat as needed.