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.

Adding Positive and Negative Numbers

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)

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8)  (8 + 5) × 35/5 + 6

9)

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10)

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Directed numbers

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

Number line

Adding and subtracting directed numbers

The rules for adding and subtracting directed numbers are:

plus minus numbers

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.

HCF

Multiples

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

LCF

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.

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

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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   Equation: 12{x^2} - 27{y^2}

Adding and subtracting algebraic fractions

Calculate Equation: frac{2}{5} + frac{3}{7}

Calculate Equation: frac{2}{3} - frac{y}{{18}}

Calculate Equation: frac{x}{y} + frac{y}{x}

Calculate Equation: frac{2}{x} - frac{5}{{x + 2}}

Calculate Equation: frac{4}{5} times frac{5}{6}

Calculate Equation: frac{5}{6} times frac{3}{8}

Calculate Equation: frac{2}{3} times frac{5}{y}

Calculate Equation: frac{{y - 5}}{{y + 1}} times frac{{y + 1}}{{y + 2}}

Calculate Equation: frac{3}{8} div frac{3}{4}

Calculate Equation: frac{9}{z} div frac{{(z + 1)}}{{(z - 8)}}

Equation of a straight line

The general equation of a straight line is Equation: y = mx + c, where Equation: m is the gradient and Equation: (0,c) 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 Equation: 2x + y - 13 = 0.

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

Equation: 2y - 5x = 12.

Function notation

The equation of a straight line can also be written in the form Equation: f(x) = mx + c.

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

Therefore, in general Equation: y = f(x).

Example:

If Equation: f(x)=4x+1, calculate Equation: f(3)

Now try the example question below.

Answer

Equation: f(3) means you substitute Equation: x=3 into Equation: 4x+1

Equation: f(3)=4(3)+1

Equation: =12+1

Equation: 13

Problem 5   If Equation: f(x) = 3x - 5, calculate Equation: f( - 2).

 

Solving equations and inequalities

Remember when solving equations and inequations to use the rule:

Change side, change operation

Example:

Solve the equation Equation: - 2 - 3y = 11

Answer:

Equation: - 3y = 11 + 2 ;

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

Equation: - 3y = 13

Equation: y = frac{{13}}{{ - 3}},or,y =  - 4frac{1}{3}

Problem 1: Solve the equation Equation: 35 = 5 - 6m to find m.

Problem 2 : Solve the equation Equation: 4(5 - 2x) - 5 = 39

Problem 3: Solve the equation Equation: 4x - 2(1 - 3x) = 3

Problem 4 : Solve the equation Equation: 3(7 - 2x) = 3(x + 5) - 84

Problem 5: Solve the equation: Equation: 6x + 15 = 4x + 17

Problem 6: Solve the equation: Equation: 3(8 - x) + 1 = x + 5

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 Equation: 3x + 2 le  - 4

Answer:

Equation: 3x + 2 le  - 4

Equation: 3x le  - 4 - 2

Equation: 3x le  - 6

Equation: x le frac{{ - 6}}{3}

Equation: x le  - 2

Problem 1: Solve the inequation Equation: 2x-1,textgreater 9

Problem 2: Solve the inequation Equation: 5(w - 1) - 8w ge  - 11

Problem 3: Solve the inequality: Equation: 5y - 16 ge y + 32

Problem 4: Solve the inequality 2x + 1 > 5

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