# Algebra

## Order of Operations

Rule 1: First perform any calculations inside brackets.

Rule 2: Next perform all multiplications and divisions, working from left to right.

Rule 3: Lastly, perform all additions and subtractions, working from left to right.

Rule 4: If  brackets are enclosed in other parenthesis, work from the inside out.

1)   (−1) + 5

2)   (−8) + (−1)

3)   11 + (−2)

4)   10 + (−11) + 5 + (−5)

5)   (−2.1) + (−1) + (−7.6)

6)  (- $\frac{1}{3}$) + (- $\frac{3}{5}$)

7)   (- $\frac{1}{2}$) + (- $\frac{5}{3}$)

8)   (- $\frac{1}{4}$) + (- $\frac{3}{2}$)

9)   ($\frac{8}{5}$) + (- $\frac{1}{3}$)

10)   0.85 + (−2.4) + 4.5

11)  (- $\frac{7}{4}$) – (- $\frac{1}{2}$)

Calculate:

1)  9 − 32 ÷ 4

2)  5(10 − 1)

3)  72 ÷ 9 + 7

4)  40 ÷ 4 − (5 − 3)

5)  (5 + 16) ÷ 7 − 2

6)  (6 − 4) × 49 ÷ 7

7)

8)  (8 + 5) × 35/5 + 6

9)

10)

## Directed numbers

The positive and negative whole numbers are called integers. They can be shown on a number line.

The rules for adding and subtracting directed numbers are:

Multiplying and dividing directed numbers

If the two signs are the same, the answer will be positive.

If the two signs are different, the answer will be positive.

Multiplication:                                         Division

(+) x (+) = +                                                  (+) ÷ (+) = +

(+) x (-) = –                                                    (+) ÷ (-) = –

(-) x (+) = –                                                    (-) ÷ (+) = –

(-) x (-) = +                                                    (-) ÷ (-) = +

Multiplying and Dividing Positives and Negatives

Work out:

1)  12 ÷ −3

2)  −120 ÷ −20

3)  −8 × −11

4)  −3 × 6 × −6

5) (-4)³ x (-1)³

6) $\frac{-6}{-3}$

7) $(-2)^{5} (-10)^{2}$

8) $(-1)^{15} (-1)^{24}$

## Multiples, factors, primes, squares and cubes

Factors

The whole numbers that divide exactly into 15 are called factors of 15. The factors of 15 are 1, 3, 5 and 15.

Multiples

The multiples of 6 are the numbers 6, 12, 18, 24, 30 …

Exercise 1: Write down all the factors of

A) 80     B) 64     C) 125     D) 90   E) 60

Exercise 2: Find the common factors of

A) 12 and 30    B) 80 and 100     C) 42 and 48    D) 6, 12 and 42

Exercise 3: Find the highest common factor (HCF) of:

A)36 and 45   B)20 and 24   C)90 and 18     D) 8, 32 and 44

Exercise 4: Find the lowest common factor (LCF) of:

A) 5 and 15   B) 6 and 9    C) 14 and 21    D) 11 and 5     E) 8, 10 and 12

Primes

A prime number is a number that has exactly two factors. 5 is a prime number because it has exactly two factors (1 and 5).

The first 10 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29

A prime factor is a factor that is also a prime number.

Example: List the prime factors of 30.

The factors of 30 are: 1, 2, 3, 5, 6, 10, 15 and 30. The prime factors of 30 are: 2,3 and 5.

Any number can be written as the product of prime factors.  For example 120 = 2³ x 3 x 5. The next example shows how a factor tree can be used.

Expressing numbers as the product of prime factors can help you to find highest common factors (HCF) and lowest common multiples (LCM).

Exercise 1: Write each of the numbers as the product of prime factors:

A) 74      B) 100    C) 98      D) 275      C) 1110     D) 2004

Exercise 2: Find the HCF and LCM of

A) 24 and 42    B) 70 and 42     C) 30, 36 and 42     D) 28, 35 and 56

The numbers 1, 4, 9 and 16 are called square numbers.

The numbers 1, 8 and 27 are called cube numbers.

### The basic properties

The commutative property:

a + b = b + a        Commutative property of addition

a · b = b · a          Commutative property of multiplication

The associative property:

a + (b + c) = (a + b) + c       Associative property of addition

a(b · c) = (a · b)c                  Associative property of multiplication

a(b + c – d) =  a · b + a · c – a · d                Distributing multiplication over  addition and                                                                               subtraction

### Identities

The numbers 0 and 1 have special roles in algebra — as identities.

a + 0 = 0 + a = a      Adding 0 to a number doesn’t change that number; the number keeps                                    its identity.

a · 1 = 1 · a = a       Multiplying a number by 1 doesn’t change that number; the number                                       keeps its identity.

### Inverses

A number and its additive inverse add up to 0 :                                9 + (–9) = 0

A number and its multiplicative inverse have a product of 1:        7.  (1/7) = 1

Any number raised to the power of “one” equals itself. One raised to any power is one.

Product Rule:  when multiplying two powers that have the same base, you can add the exponents.

Power Rule: to raise a power to a power, just multiply the exponents.

Quotient Rule:  we can divide two powers with the same base by subtracting the exponents.

Any nonzero number raised to the power of zero equals 1:

Negative Exponents: any nonzero number raised to a negative power equals its reciprocal raised to the opposite positive power:

### Evaluating Expressions

1.    $p^{2}$+ m; use m = 1, and p = 5
2.    y + 9 − x; use x = 1, and y = 3
3.    $p^{3}$m ÷ 4; use m = 4, and p = 7
4.    c × bc/4 − (7 − a); use a = 4, b = 8, and c = 5
5.     $(p + q)^{2}$ − (5 − 5); use p = 1, and q = 1

### Simplify each expression

1)  4(1 + 9x)

2)  -8(−4 − 3n)

3)  −6(x + 4)

4)  −4(−8x − 8)

5)  −8(1 − 5x)

6) 3b – (4b – 6b + 2) + b

7) $\frac{36p^{5}q^{4}}{9p^{3}q}$

8)  -5(2-x) – 3(5x-4) –  (x²)²

9) -2(m-5) – 4 +m

10) 3(x+y) – 2(3x-y)

### Factorise:

1.  6a – 9
2.  x² +5x
3.  3a² -12a
4.  20xy – 6x
5. 20y² – 5y
6. a² – b²   hint: proof that a² – b² = (a-b)(a+b)
7. 25p² -1
8. 16 – w²
9.  y² -81
10.  3a² -27

Calculate by factorising  14² – 9²

Factorise fully

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### Equation of a straight line

The general equation of a straight line is , where  is the gradient and  the coordinates of the y-intercept.

Problem 1 Find the equation of the line with gradient 3, passing through (4, 1).

Problem 2 Find the equation of the line which passes through the points A(-2, 0) and B(1, 6) and state the gradient and y-intercept.

Problem 3 Find the gradient and y-intercept for the straight line with equation .

Problem 4 Find the gradient and y-intercept for the straight line with equation:

.

#### Function notation

The equation of a straight line can also be written in the form .

This is called function notation and is just another way of expressing the relationship between two variables.

Therefore, in general .

Example:

If , calculate

Now try the example question below.

means you substitute  into

Problem 5   If , calculate .

### Solving equations and inequalities

Remember when solving equations and inequations to use the rule:

Change side, change operation

Example:

Solve the equation

;

Move the -2 over to the right hand side changes it to +2

-3 is multiplying on the left, so when moved to the right it divides

Problem 1: Solve the equation  to find m.

Problem 2 : Solve the equation

Problem 3: Solve the equation

Problem 4 : Solve the equation

Problem 5: Solve the equation:

Problem 6: Solve the equation:

Problem 7: Solve the equation: 3j + 2 = 5j – 4

Problem 8: Solve the equation:  11 – 2(1-v) = 3 – 4v

#### Inequalities

When solving inequations, the inequality sign can mostly be treated the same way as an equals sign.

Example: Solve the inequation