ALGEBRAExpressions and

Equations

- Linear number patterns are sequences of numbers where the difference between terms is always the same (constant) - Rule generating a linear pattern is: Difference × n ± a constant

e.g. Write a rule (using n) to describe the following number patterns.

n Number of Squares (s)

Number of Dots

(d)

1 1 5

2 4 9

3 7 13

4 10 17

5 13 21

+ 3+ 3+ 3+ 3

+ 4

+ 4

+ 4

+ 4

Rule: s =

Rule: d =3×n 4×n

3×1= 33 = 1- 2

- 2

4×1= 44 = 5+ 1

+ 11. Find the difference between terms and if the same multiply by n

2. Substitute to find constant3. Check if rule works

3×4 – 2

4×4 + 1

FINDING RULES FOR LINEAR PATTERNS

CREATING EXPRESSIONS

Note: × and ÷ are not often used in Algebra

i.e. 5 × x = 5x i.e. 8 ÷ x = 8

x

Also a dot ‘.’ means multiply i.e. 2x . 2y = 2x × 2y

- Using suitable symbols to express rules

e.g. Write an expression for each of the following

a) A number with 12 added to it

b) A number with 9 subtracted from it

c) A number multiplied by 2

d) A number divided by 6

As long as you explain what a symbol represents, any symbol can be used

Let n = a number

n + 12

n - 9

n × 2 Best written as 2n

n ÷ 6 Best written as n 6

e.g. John has x dollars. How much will he have if:

a) He spends $35

b) He is given $28

c) He doubles his money

d) He spends half

x - 35

x + 28

x 2

x2

Once you have an expression, it can be used to calculate values if you know what the ‘variable’ (symbol) is worth.

e.g. John has $50, use the expressions to calculate

how much he will have in each situation:

50

- 35

50

+ 28

x 50 2

50 2

= $15

= $78

= $100

= $25

- ALL terms can be multipliedRules: 1) Multiply all numbers in the expression

2) Place letters in alphabetical order behind producte.g. Simplify:

a) 4c × 2d

b) –p × -2q × -6r

No number = 1 i.e. -p = -1p

= 4 × 2 × c × d= 8cd

= -1 × -2 × -6 × p × q × r= -12pqr

SIMPLIFYING EXPRESSIONS BY MULTIPLYING

POWERS- Remember: 3 × 3 × 3 ×

3 = 34

- Variables (letters) that are multiplied by themselves are treated the same way

e.g. Simplify these expressions that are written in full

a) r × r b) p × p × p × p × p

= r2 = p5

- Sometimes there may be two or more variables

e.g. Simplify

a) a × a × b × b × b

b) d × e × e × d × f

= a2b3 = d2e2fLetters should still be written in alphabetical order!

FOUR POWER RULES1. Multiplication

- Does x2 × x3 = x × x × x × x × x ?

YES- Therefore x2 × x3 = x5

- How do you get 2 3 = 5 ?+

- When multiplying index (power) expressions with the same letter, ADD the powers.

e.g. Simplify

a) p10 × p2 b) a3 × a2 × a= p(10 + 2) = a(3 + 2 + 1)

= p12 1

No number = 1 i.e. p = 1p1

= a6

- Remember to multiply any numbers in front of the variables first

e.g. Simplify

a) 2x3 × 3x4 b) 2a2 × 3a × 5a4

= 2 × 3 = 2 × 3 ×

5= 6

1

= 30

x(3 + 4)

a(2 + 1 + 4)x7

a7

2. Division

YES- Does 6 = 1 ? 6

- Therefore x = 1 x- Does x5 = x × x × x × x

× x ? x3 x × x × x

YES = x × x × 1 × 1 × 1

- Therefore x5 = x2

x3

- When dividing index (power) expressions with the same letter, SUBTRACT the powers.

e.g. Simplify

a) p5 ÷ p = p(5 - 1) = x(7 - 4)

= p4

= x3

- Remember to divide any numbers in front of the variables first

e.g. Simplify

a) 12x5 ÷ 6x4

= 12 ÷ 6= 2

- How do you get 5 3 = 2 ?-

1 b) x7

x4

If the power remaining is 1, it can be left out of the answer

x(5 - 4) b) 5a7

15a2

÷ 5÷ 5

= 1 5

a(7 - 2)

= 1 5

xa5 or a5

5

3. Powers of powers

- Does (x2)3 = x2 × x2 × x2 ? YES

- Therefore (x2)3 = x6

- How do you get 2 3 = 6 ?×

- Does x2 × x2 × x2 = x6 ? YES

- When taking a power of an index expression to a power, MULTIPLY the powers

e.g. Simplify

a) (c4)6 b) (a3)3

= c(4 × 6) = a(3 × 3)

= c24 = a9

- If there is a number in front, it must be raised to the power, not multipliede.g. Simplify

a) (3d2)3 b) (2a3)4× d(2 × 3)

= 27 a12= 33 × a(3 × 4)= 24

d6 = 16- If there is more than one term in the brackets, raise all to the power

e.g. Simplify

a) (x3y z4)3

= x(3 × 3)

= x9

1 y(1 × 3)z(4 × 3)

y3z12

b) (4b2c5)2 b(2 × 2)

= 16

c(5 × 2)

b4c10

= 42

4. Powers of zero- Any base to the power of zero has a value of 1

e.g. x0 = 1SQUARE ROOTS

- Simply halve the power (as √x is the same as x½ )

e.g. Simplify

a) = 10x4

b)= 8x3

LIKE AND UNLIKE TERMS- LIKE terms are those with exactly the same letter, or

combination of letters and powers

LIKE terms: UNLIKE terms:

2x, 3x, 31x4ab, 7ab

2x, 35x, 6x2

2ab, 2ace.g. Circle the LIKE terms in the following groups:

a) 3a 5b 6a 2c b) 2xy 4x 12xy 3z 4yx

While letters should be in order, terms are still LIKE if they are not.

100x8

64x6= 100 x

8 = 64 x6

SIMPLIFYING BY ADDING/SUBTRACTING- We ALWAYS aim to simplify expressions from expanded to

compact form- Only LIKE terms can be added or subtracted- When adding/subtracting just deal with the numbers in front of the

letterse.g. Simplify these expanded expressions into compact form:

a) a + a + a

b) 5x + 6x + 2x

c) 3p + 7q + 2p + 5q

= (1 + 1 + 1)a= 3a

= (5 + 6 + 2)x= 13x

= (3 + 2)p= 5p + 12q

d) 4a + 3a2 + 7a + a2

1 1 1

(+ 7 + 5)q

1 = (4 + 7)a

(+ 3 + 1)a2= 11a + 4a2

- For expressions involving both addition and subtraction take note of signs

a) 4x + 2y – 3x

b) 3a – 4b – 6a + 9b

c) 3x2 - 9x + 6x2 + 8x - 5

= (4 – 3)x= x + 2y = (3 -

6)a= -3a + 5b

= 9x2 - x - 5

d) 4ab2 +2a2b – 5ab2 + 3ab

(- 9 + 8)x

= -ab2 + 2a2b + 3ab

+ 2y(- 4 + 9)b

= (3 + 6)x2 - 5

If the number left in front of a letter is 1, it can be left out

e.g. Simplify the following expressions:

EXPANDING EXPRESSIONS- Does 6 × (3 + 5) = 6 × 3 + 6

× 5 ? 6 × 8 = 18 + 30 48 = 48

YES

- The removal of the brackets is known as the distributive law and can also be applied to algebraic expressions

- When expanding, simply multiply each term inside the bracket by the term directly in front

e.g. Expand

a) 6(x + y) b) -4(x – y)

c) -4(x – 6) d) 7(3x – 2)

e) x(2x + 3y) f) -3x(2x – 5)

= 6 × x

+ 6 × y =

6x

= -4 × x

- -4 × y = -4x

= -4 × x - -4 × 6 = -4x

= 7 × 3x

- 7 × 2 = 21x

= x × 2x

+ x × 3y =

2x2

1 1

= -3x × 2x

- -3x × 5 = -

6x2 Don’t forget to watch for sign changes!

+ 6y + 4y

+ 24 - 14

+ 3xy + 15x

- If there is more than one set of brackets, expand them all then collect any like terms.

e.g. Expand and simplify

a) 2(4x + y) + 8(3x – 2y)

b) -3(2a – 3b) – 4(5a + b)

= 2 × 4x

+ 2 × y

+ 8 × 3x

- 8 × 2y =

8x + 2y +

24x - 16y

= 32x - 14y

= -3 × 2a - -3 × 3b

- 4 × 5a

+ -4 × 1b = -

6a + 9b - 20a - 4b

= -26a

+ 5b

SUBSTITUTION- Involves replacing variables with numbers and calculating the

answer- Remember the BEDMAS rules

e.g. If m = 5, calculate m2 – 4m - 3

= 52 – 4×5 - 3= 25 – 4×5 - 3= 25 – 20 - 3= 2

- Formulas can also have more than one variable

e.g. If x = 4 and y = 6, calculate 3x – 2y

= 3×4 - 2×6= 12 - 12= 0

e.g. If a = 2, and b = 5, calculate 2b – a 4

Because the top needs to be calculated first, brackets are implied

= (2 × 5 – 2) 4= (10 – 2) 4= 8 4

= 2

ALGEBRAIC FRACTIONS

- Are fractions with letters in them. - Should be treated exactly like normal fractions.

1. Simplifying- As with normal fractions, look for common numbers and letter to

cancel out. Do numbers first, then letters

e.g. Simplify:

a) 3a 6a

÷ 3÷ 3

= 1 2

1

1

a(1 – 1)

= 1 2

b) 18x 6y

÷ 6÷ 6

= 3 1

xy

= 3x y

c) 4x3

6x2

÷ 2÷ 2

= 2 3

x(3 – 2)

= 2 3

x

2. Multiplying Fractions- Multiply top and bottom terms separately then simplify.

a) b × b2

2 5

e.g. Simplify:

= b3

10

b) y2 × 4 3 y

= 4y2

3y= 4y 33. Dividing Fractions

- Multiply the first fraction by the reciprocal of the second, then simplify

Note: b is the reciprocal of 2 2 b

a) 2a ÷ a2

5 3

e.g. Simplify:

= 6a 5a2

= 6 5a

or 6a-1

5

= b × b2

2 × 5= y2 × 4

3 × y

= 2a 5

× 3a2

= 2a × 35 × a2

4. Adding/Subtracting Fractions a) With the same denominator:

- Add/subtract the numerators and leave the denominator unchanged. Simplify if possible.

a) 3x + 3x

10 10

e.g. Simplify:= 3x +

3x 10= 6x 10÷ 2÷ 2

= 3x 5

b) 6a - b 5 5

= 6a - b 5

b) With different denominators: - Multiply denominators to find a common

term. - Cross multiply to find equivalent numerators. - Add/subtract fractions then simplify.

e.g. Simplify:a) a +

2a 2 3= 3a + 4a

6= 7a 6

b) 2x – 5x

3 4= 8x – 15x 12= -7x 12

= 2×3

3×a + 2×2a = 3×4

4×2x - 3×5x

FACTORISING EXPRESSIONS- Factorising is the reverse of expanding- To factorise: 1) Look for a common factor to put outside the

brackets2) Inside brackets place numbers/letters needed to make up original terms

e.g. Factorise

a) 2x + 2y b) 2a + 4b – 6c

= 2( ) x + y = 2( )

a + 2b

e.g. Factorise

a) 6x - 15 b) 30a + 20

= 3( ) 2x

- 5 = 10( ) 3a + 2

- Always look for the highest common factor

You should always check your answer by expanding it

e.g. Factorise

a) 6x + 3 b) 20b - 10= 3( ) 2x

+ 1 = 10( ) 2b - 1

- Sometimes a ‘1’ will need to be left in the brackets

- 3c

e.g. Factorise

a) cd - ce b) xyz + 2xy – 3yz = c( ) d - e = y( )

xz + 2x

- Letters can also be common factors

e.g. Factorise

a) 5a2 – 7a5 b) 4b2 + 6b3= a2( ) 5 - 7a3 = 2b2( ) 2 + 3b

- Powers greater than 1 can also be common factors

- 3z

c) 4ad – 8a

= 4 ( ) a

d

- 2

SOLVING EQUATIONS- When solving we need to isolate the unknown variable to find its

value- To isolate we work backwards by undoing operations

1) To undo multiplication we use divisione.g. Solve 3x = 18

÷3 ÷3x = 6

2) To undo addition we use subtractione.g. Solve x + 2 = 6

-2 -2x = 4

3) To undo subtraction we use additione.g. Solve x - 8 = 11

+8 +8x = 194) To undo division we use

multiplication

×5x = 30

e.g. Solve x = 6 5

×5

- Terms containing the variable (x) should be placed on one side (often left)e.g. Solve

a) 5x = 3x + 6

b) -6x = -2x + 12-3x -3x

2x = 6÷2÷2

x = 3

You should always check your answer by substituting into original equation

- Does 5×3 = 3×3 + 6 ?

15 = 9 + 6

YES

+2x

+2x -4x = 12

÷-4

÷-4 x = -3

Always line up equals signs and each line should contain the variable and one equals sign

- Does -6×-3 = -2×-3 + 12 ?

18 = 6 + 12

YES

- Numbers should be placed on the side opposite to the variables (often right)e.g. Solve

a) 6x – 5 = 13

b) -3x + 10 = 31+5 +5

6x = 18÷6÷6

x = 3

-10 -10 -3x = 21

÷-3

÷-3 x = -7

Always look at the sign in front of the term/number to decide operation

Don’t forget the integer

rules!

- Same rules apply for combined equations

e.g. Solvea) 5x + 8 = 2x + 20

b) 4x - 12 = -2x + 24-2x -2x

3x + 8 = 20 -8-8

3x = 12

+2x

+2x 6x - 12 =

24

÷6÷6 x = 6

÷3÷3 x = 4

+12

+12 6x = 36

- Answers can also be negatives and/or fractions

e.g. Solvea) 8x + 3 = -12x - 17

b) 5x + 2 = 3x + 1+12

x+12x 20x + 3 = -

17 -3-3 20x = -20

-3x -3x 2x + 2 = 1

÷2÷2 x = -1 2

÷20

÷20 x = -1

-2-2 2x = -1

Make sure you don’t forget to leave the sign

too!

Answer can be written as a decimal but easiest to leave as a

fraction

- Expand any brackets first

e.g. Solve

a) 3(x + 1) = 6 b) 2(3x – 1) = x + 8

-3-3 3x = 3

÷3÷3 x = 1

3x+ 3= 6 6x- 2 = x + 8 -x-x

5x - 2 = 8 +2+2

5x = 10÷5÷5

x = 2- For fractions, cross multiply, then solve

e.g. Solve

a) x = 9

4 2

2x= 36÷2÷2

x = 18

b) 3x - 1 = x + 3

5 22(3x - 1)= 5(x +

3)6x

- 2 = 5x+ 15-5x-5x

x - 2 = 15 +2+2

x = 17

- For two or more fractions, find a common denominator, multiply it by each term, then solve

e.g. Solve 4x - 2x = 10

5 3

5 × 3 = 15

×15

×15

×15

60x

5

- 30x

3

= 150

Simplify terms by dividing numerator by denominator12

x

- 10x

= 150

2x = 150 ÷2÷2

x = 75

WRITING EQUATIONS AND SOLVING- Involves writing an equation and then solvinge.g. Write an equation for the following information

a) I think of a number, multiply it by 3 and then add 12. The result is 36.

a) I think of a number, multiply it by 5 and then subtract 4. The result is

n3 + 12

= 36

Let n = a number

Let n = a number

n5 - 4 = n + 18

the same as if 18 were added to the number

e.g. Write an equation for the following information and solvea) A rectangular pool has a length 5m longer than its width. The

perimeter of the pool is 58m. Find its width

Draw a diagram

Let x = width

-10-10

Therefore width is 12 m

x x

x + 5

x + 5

x + 5 + x + x + 5 + x = 58 4x + 10 =

58 4x = 48

÷4÷4x = 12

b) I think of a number and multiply it by 7. The result is the same as if I multiply this number by 4 and add 15. What is this number?

Let n = a number

n = n

+ 15-4n-4n

3n = 15

Therefore the number is 5

7 4

÷3÷3n = 5

- Inequations contain one of four inequality signs: < > ≤ ≥ - To solve follow the same rules as when solving equations- Except: Reverse the direction of the sign when dividing by a negative

e.g. Solvea) 3x + 8 > 24

b) -2x - 5 ≤ 13-8-8

3x > 16 ÷3÷3x > 16 3

+5+5-2x ≤ 18

÷-2

÷-2

Sign reverses as dividing by a negative

x ≥ -9

As answer not a whole number, leave as a fraction

SOLVING INEQUATIONS

CHANGING THE SUBJECT- Involves rearranging the formula in order to isolate the new ‘subject’- Same rules as for solving are

usede.g.

a) Make x the subject of y = 6x - 2 +2+2

y + 2 = 6x÷6÷6

y + 2 = x 6

All terms on the left must be divided by 6

b) Make R the subject of IR = V÷I÷I

Treat letters the same as numbers!R = V

Ic) Make x the subject of y = 2x2

÷2÷2y = x2

2

Taking the square root undoes squaringy

2=x

Remember: When rearranging or changing the subject you are NOT finding a numerical answer