If you work through this section you should be able to:

- Understand the key definitions and methods used in algebra.
- Understand how algebra can be used to simplify and represent formulas.

You may also find the Algebraic expressions and Equations sections useful.

You may find the Mathematical operations section useful if you need to refresh your knowledge of the sequence of mathematical calculations, and of calculations with negative numbers, before looking at algebra.

We use algebra to abbreviate information without loss of clarity or accuracy, for example to represent a formula, and to generalise rules and relationships by using letters to represent values.

Letters are most often in lower-case and it is conventional to use the letters early in the alphabet (a, b, c…) to stand for constant values, and those at the end of the alphabet (such as x, y and z) to represent variable quantities or as the unknowns in equations.

We combine letters and numbers and more complex terms and expressions using operators (+, −, ×, ÷, raising to a power, etc.) and these follow the normal rules of arithmetic with a few extra conventions.

We do not usually use the × sign explicitly: it is understood that 3p means 3 × p, xy means x × y, and 2(a + b) means 2 × (a + b).

Also we rarely use the ÷ sign but show division in fraction form. So a ÷ b is written as $\frac{a}{b}$.

The rest of this section will look at some basic definitions and processes of algebra.

A term is a unit in an algebraic expression. A term can be:

- A pure number (2, 3, 12),
- A single letter, which may be preceded by a number (a, 2x, 4b),
- A combination of letters (4xy, 3abc, pq).

Some or all letters may be raised to a power(x^{3}, 4a^{2}b, 5p^{2}q^{3}).

If an expression has more than one term they will be separated by + or − signs.

The coefficient is the number in front of a letter or group of letters ("how many x's"). By convention we usually drop a coefficient of 1 so 1x is written as simply x, and a^{2}b means 1a^{2}b.

A letter is raised to a power to show how many of that letter are to be multiplied together. The word exponent is sometimes used instead of power.

So a^{2} = a × a; a^{5} = a × a × a × a × a; and of course a^{1} = a.

a^{0} = 1 for any value of a

To multiply letters raised to powers, **add** the powers.

To divide letters raised to powers, **subtract **the powers.

a^{2} × a^{3} × a = a^{2+3+1} = a^{6}

a^{7} ÷ a^{3} = a^{7-3} = a^{4}

x^{4} ÷ x^{3} = x^{1} = x

Note that the same letter must be involved in this combining of powers. Terms of mixed letters can be operated on though, as in the following example.

3a^{2}b^{4}c × 7a^{3}b^{5}c

= 3 × 7 × a^{2} × a^{3} × b^{4} × b^{5} × c^{1} × c^{1}

= 21a^{5}b^{9}c^{2}

Note: Different powers of the same letter (a, a^{2}, a^{3}…) form **unlike** terms (see next page for like and unlike terms) so they cannot be simply added or subtracted.

If a term that involves letters raised to powers is itself raised to a power then the powers are **multiplied**. Note: any coefficient (pure number) is raised to the power as normal.

(x^{3})^{2} = x^{3×2} = x^{6}

(2y^{2})^{3} = 2^{3} × y^{2×3} = 8y^{6}

(3a^{4}b^{5})^{2} = 3^{2}a^{4×2}b^{5×2} = 9a^{8}b^{10}

**Like terms** are terms with the same letter or combination of letters, although the coefficients may be different. For example, 4wp, wp and 5wp are like terms.

All pure numbers (no letters) are like terms too. So 3, 7 and 23 are like terms.

Some sets of like terms are:

1, 9, 53…

a, 3a, 7a…

5x^{2}, 2x^{2}, x^{2}…

ab, 5ab, 2ab…

3pq^{2}, 12pq^{2}, pq^{2}…

Each corresponding letter in a set of like terms must be raised to the same power, so 3x, 2x^{2} and x^{3} are **unlike terms**. So are ab, a^{2}b, ab^{2}, a^{2}b^{2}.

Expressions are combinations of terms linked by plus or minus signs.

More complex expressions may also include simpler expressions in brackets.

Some expressions are:

3x + 4y − 2xy

2x^{2} + 3x + 7

5 − 2 (x − 3)^{2}

(2a − 3b)(3a + 2b)

When we want to simplify an expression, we combine like terms together. This essentially means adding and subtracting the coefficients of the like terms.

4w + 3 + 2w − 8

= 6w −5

2ab + 3ac +4ab − ac

= 6ab +2ac

3x^{2} − 2x + 4 − 2x^{2} + 7x − 1

= x^{2} + 5x + 3

**1)** w + w

**2)** y + y + y + y

**3)** x + x + x

**4) **w + 3w + 5w

**5)** 9y − y − 7y

**6) **a + 2a − 5a

**7)** w^{2} + 2w^{2} + 3w^{2}

**8)** wy × 4z + 2w × yz + 3y × wz

**9) **x + 2y − 2xy + 3y − 3x + 5xy

and (where possible) the following:

**10)** w × 3w − 2w × w + w + 2w

**11)** 4wy − 2yz + 7wy − 6yz

**12)** 2a + 13b − 12ab + 5c − 3ac

**13)** 1 + w + 2w^{2} − 3 − 2w + 5w^{2}

**14)** a^{2} + b^{2} + c^{2} − 3a^{2} + 4b^{2} + 2c^{2}

**15)** 3abc − 2bc + 3ac + 4bac − 5ca + 4cb

**16)** x^{2} + 3x^{3} − 4x + 2x^{3} − 4x^{2} + 5x

**17) **2ab + 3bc + 4ac − 5abc

**18)** 8a × 2cb − 3b × 3ca + 5c × 2ba

**19)** 5ax + 7by + 6cz − 2xa − 5zc + 3by

**1)** w + w

**2w**

**2)** y + y + y + y

**4y**

**3)** x + x + x

**3x**

**4) **w + 3w + 5w

**9w**

**5)** 9y − y − 7y

**y**

**6) **a + 2a − 5a

**−2a**

**7)** w^{2} + 2w^{2} + 3w^{2}

**6w ^{2}**

**8)** wy × 4z + 2w × yz + 3y × wz

**9wyz**

**9) **x + 2y − 2xy + 3y − 3x + 5xy

**−2x + 5y + 3xy**

and (where possible) the following:

**10)** w × 3w − 2w × w + w + 2w

**w ^{2} + 3w**

**11)** 4wy − 2yz + 7wy − 6yz

**11wy − 8yz**

**12)** 2a + 13b − 12ab + 5c − 3ac

**2a + 13b − 12ab + 5c − 3ac (already in simplest form)**

**13)** 1 + w + 2w^{2} − 3 − 2w + 5w^{2}

**−2 − w + 7w ^{2}**

**14)** a^{2} + b^{2} + c^{2} − 3a^{2} + 4b^{2} + 2c^{2}

**−2a ^{2} + 5b^{2} + 3c^{2}**

**15)** 3abc − 2bc + 3ac + 4bac − 5ca + 4cb

**7abc + 2bc − 2ac**

**16)** x^{2} + 3x^{3} − 4x + 2x^{3} − 4x^{2} + 5x

**−3x ^{2} + x + 5x^{3}**

**17) **2ab + 3bc + 4ac − 5abc

**2ab + 3bc + 4ac − 5abc (already in simplest form)**

**18)** 8a × 2cb − 3b × 3ca + 5c × 2ba

**17abc**

**19)** 5ax + 7by + 6cz − 2xa − 5zc + 3by

**3ax + 10by + cz**

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