Learning Outcomes

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.
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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 $ab$.

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

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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(x3, 4a2b, 5p2q3).

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 a2b means 1a2b.

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 a2 = a × a; a5 = a × a × a × a × a; and of course a1 = a.

a0 = 1 for any value of a

To multiply letters raised to powers, add the powers.

To divide letters raised to powers, subtract the powers.

Example

a2 × a3 × a = a2+3+1 = a6

a7 ÷ a3 = a7-3 = a4

x4 ÷ x3 = x1 = 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.

Examples

3a2b4c × 7a3b5c

= 3 × 7 × a2 × a3 × b4 × b5 × c1 × c1

= 21a5b9c2

Note: Different powers of the same letter (a, a2, a3…) 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.

Examples

(x3)2 = x3×2 = x6

(2y2)3 = 23 × y2×3 = 8y6

(3a4b5)2 = 32a4×2b5×2 = 9a8b10

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.

Examples

Some sets of like terms are:

1, 9, 53…

a, 3a, 7a…

5x2, 2x2, x2

ab, 5ab, 2ab…

3pq2, 12pq2, pq2

Each corresponding letter in a set of like terms must be raised to the same power, so 3x, 2x2 and x3 are unlike terms. So are ab, a2b, ab2, a2b2.

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

More complex expressions may also include simpler expressions in brackets.

Examples

Some expressions are:

3x + 4y − 2xy

2x2 + 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.

Examples

4w + 3 + 2w − 8
= 6w −5

2ab + 3ac +4ab − ac
= 6ab +2ac

3x2 − 2x + 4 − 2x2 + 7x − 1
= x2 + 5x + 3

Simplify the following:

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) w2 + 2w2 + 3w2

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 + 2w2 − 3 − 2w + 5w2

14) a2 + b2 + c2 − 3a2 + 4b2 + 2c2

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

16) x2 + 3x3 − 4x + 2x3 − 4x2 + 5x

17) 2ab + 3bc + 4ac − 5abc

18) 8a × 2cb − 3b × 3ca + 5c × 2ba

19) 5ax + 7by + 6cz − 2xa − 5zc + 3by

Simplify the following:

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) w2 + 2w2 + 3w2

6w2

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

w2 + 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 + 2w2 − 3 − 2w + 5w2

−2 − w + 7w2

14) a2 + b2 + c2 − 3a2 + 4b2 + 2c2

−2a2 + 5b2 + 3c2

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

7abc + 2bc − 2ac

16) x2 + 3x3 − 4x + 2x3 − 4x2 + 5x

−3x2 + x + 5x3

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

Activity

You can download a version of this Introduction to algebra activity in Word format: