Simplifying Algebra

What you'll learn

What letters, terms and coefficients mean in algebra.
How to collect like terms, such as $x$ terms with other $x$ terms.
How to simplify algebra with multiplication, division and powers.
How to avoid mixing up terms like $x$ , $x^2$ and $x y$ .

The basic idea

Simplifying algebra means writing an expression in a shorter, neater way without changing its value.

For example, if you have three lots of $p$ , you can write that as $3 p$ .

Definition

Useful algebra words

A variable is a letter that stands for a number, such as $x$ , $a$ or $m$ .
An expression is a collection of numbers, letters and operations, such as $4 x + 3 x$ .
A term is one part of an expression, such as $4 x$ or $- 2 a$ .
A coefficient is the number multiplying the variable. In $5 x$ , the coefficient is 5.

Key Idea

Simplifying keeps the meaning the same

You are not solving for the letter. You are just making the expression shorter and easier to read.

Example

Writing repeated letters as one term

Simplify $r + r + r + r$ .

Four separate lots of r can be grouped as 4r.

Each $r$ means one lot of $r$ .
There are four lots of $r$ .
So the simplified expression is $4 r$ .

Collecting like terms

Definition

Like terms

Like terms have exactly the same letter part. For example, $3 x$ , $7 x$ and $- 2 x$ are like terms because they are all $x$ terms.

You can add or subtract like terms by adding or subtracting their coefficients.

One type of letter

Example

Collecting one-letter terms

Simplify $6 t + 3 t - 4 t$ .

Combine the coefficients of the like t terms, remembering that the last term is negative.

Look at the coefficients: 6, 3 and -4.
Combine them: 6 + 3 - 4 = 5.
Keep the same letter part, $t$ .
The answer is $5 t$ .

Common Mistake

Forgetting the minus sign

In an expression like $9 a - 2 a$ , the second term is $- 2 a$ , not just $2 a$ . The sign belongs to the term after it.

Collecting different letters

If an expression has different letters, collect each type separately.

For example, $g$ terms go with $g$ terms, and $h$ terms go with $h$ terms.

Example

Collecting two different letters

Simplify $8 g + 4 h - 3 g + 2 h$ .

The g terms combine with g terms and the h terms combine with h terms.

Find the $g$ terms: $8 g - 3 g$ .
Simplify them: $8 g - 3 g = 5 g$ .
Find the $h$ terms: $4 h + 2 h$ .
Simplify them: $4 h + 2 h = 6 h$ .
Put the simplified parts together: $5 g + 6 h$ .

Tip

Underline matching terms

In the exam, you can underline all the $x$ terms one way and all the $y$ terms another way. This helps you avoid mixing them.

Constants: numbers without letters

Definition

Constant

A constant is a number on its own, with no letter attached, such as 4 or -7.

Constants only combine with other constants. They do not combine with letter terms.

Example

Collecting letters and constants

Simplify $5 a + 6 - 2 a + 3$ .

Letter terms collect with letter terms, while constants collect only with constants.

Collect the $a$ terms: $5 a - 2 a = 3 a$ .
Collect the constants: 6 + 3 = 9.
Write both simplified parts: $3 a + 9$ .

Common Mistake

Adding unlike terms

You cannot simplify $3 a + 4 b$ into $7 ab$ . The terms have different letters, so they are not like terms.

Multiplying algebra terms

When multiplying algebra, multiply the number parts and then multiply the letter parts.

In algebra, we usually leave out the multiplication sign. So $\times b$ is written as $ab$ , and $\times x$ is written as $4 x$ .

Multiplying a number and a letter term

Example

Multiplying a number by an algebra term

Simplify $\times 4x$ .

Multiply the numbers: 5 times 4 is 20.
Keep the letter $x$ .
The answer is $20 x$ .

Multiplying two algebra terms

Example

Multiplying terms with different letters

Simplify $3p \times 6q$ .

When multiplying terms, multiply the coefficients and then join the letter parts.

Multiply the coefficients: 3 times 6 is 18.
Multiply the letters: $\times q = pq$ .
Put them together: $18 pq$ .

Common Mistake

Adding instead of multiplying

For $2a \times 5b$ , multiply 2 and 5 to get 10. The answer is $10 ab$ , not $7 ab$ .

Powers and repeated letters

Definition

Power or index

A power, also called an index, tells you how many times a letter is multiplied by itself. For example, $y^3$ means $\times y \times y$ .

So:

$\times x = x^2$
$\times m \times m = m^3$

Example

Writing repeated multiplication with a power

Simplify $\times z \times z$ .

Three repeated factors of z are written using the power z^3.

Count how many $z$ letters are being multiplied.
There are three $z$ letters.
Write this as $z^3$ .

Squared terms are different from ordinary letter terms

Terms like $x^2$ are not like terms with $x$ .

Example

Collecting squared and ordinary terms

Simplify $4x^2 - 3x + 2x^2 + 5x$ .

Squared terms and ordinary x terms are different types, so they must be collected separately.

Collect the $x^2$ terms: $4x^2 + 2x^2 = 6x^2$ .
Collect the $x$ terms: $- 3 x + 5 x = 2 x$ .
Write both parts: $6x^2 + 2x$ .

Common Mistake

Mixing $x^{2}$ and $x$

You cannot combine $6x^2 + 2x$ into $8x^2$ or $8 x$ . The letter parts are different.

Dividing algebra terms

You can divide the number part while keeping the letter part.

For example, $12h3\frac{12h}{3}$ simplifies to $4 h$ because 12 divided by 3 is 4.

Example

Dividing a simple algebra term

Simplify $15k5\frac{15k}{5}$ .

Divide the coefficient: 15 divided by 5 is 3.
Keep the letter $k$ .
The answer is $3 k$ .

Sometimes you need to simplify the top of the fraction first.

Example

Simplifying before dividing

Simplify $8m+4m4\frac{8m + 4m}{4}$ .

First simplify the numerator, then divide the whole term by 4.

First collect the terms on the top: $8 m + 4 m = 12 m$ .
Now divide by 4: $12m4=3m\frac{12m}{4} = 3m$ .
The simplified answer is $3 m$ .

Common Mistake

Cancelling letters

A letter divided by the same letter makes 1, so $6dd=6\frac{6d}{d} = 6$ . This only works when the same letter is multiplying the whole term.

Exam technique

In the exam

Circle or underline like terms before you start simplifying.
Keep the sign in front of each term, especially with subtraction.
Check that you have not combined different letter parts, such as $a$ with $b$ , or $x$ with $x^2$ .

Self review

Check yourself

Can you simplify $7 n - 2 n + n$ ?
Which terms are like terms in $4 a + 3 b - a + 5 b$ ?
What is the difference between $x + x$ and $\times x$ ?

Simplifying Algebra

What you'll learn

What letters, terms and coefficients mean in algebra.
How to collect like terms, such as $x$ terms with other $x$ terms.
How to simplify algebra with multiplication, division and powers.
How to avoid mixing up terms like $x$ , $x^2$ and $x y$ .

The basic idea

Simplifying algebra means writing an expression in a shorter, neater way without changing its value.

For example, if you have three lots of $p$ , you can write that as $3 p$ .

Definition

Useful algebra words

A variable is a letter that stands for a number, such as $x$ , $a$ or $m$ .
An expression is a collection of numbers, letters and operations, such as $4 x + 3 x$ .
A term is one part of an expression, such as $4 x$ or $- 2 a$ .
A coefficient is the number multiplying the variable. In $5 x$ , the coefficient is 5.

Key Idea

Simplifying keeps the meaning the same

You are not solving for the letter. You are just making the expression shorter and easier to read.

Example

Writing repeated letters as one term

Simplify $r + r + r + r$ .

Four separate lots of r can be grouped as 4r.

Each $r$ means one lot of $r$ .
There are four lots of $r$ .
So the simplified expression is $4 r$ .