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Expressions and Equations

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Note: × and ÷ are not often used in Algebra i.e. 5 × x = 5xi.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× 2Best written as 2n n÷ 6Best written as n 6

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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 x 2 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

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- ALL terms can be multiplied Rules: 1) Multiply all numbers in the expression 2) Place letters in alphabetical order behind product e.g. Simplify: a) p × 2qb) 2a × 5b c) 3 × 4x × 2y No number = 1 i.e. p = 1p = 1 × 2 × p × q = 2pq = 2 × 5 × a × b = 10ab = 3 × 4 × 2 × x × y = 24xy - LIKE terms are those with exactly the same letter or combination of letters LIKE terms:UNLIKE terms:2x, 3x, 31x 4ab, 7ab 2x, 3 5p, 6q 2ab, 2ac e.g. Circle the LIKE terms in the following groups: a) 3a 5b 6a 2cb) 2xy 4x 12xy 3z 4yx While letters should be in order, terms are still LIKE if they are not.

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- 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 letters e.g. Simplify these expanded expressions into compact form: a) a + a + ab) 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 + 3b + 7a + b 1 1 1 (+ 7 + 5)q1 = (4 + 7)a(+ 3 + 1)b = 11a + 4b - For expressions involving both addition and subtraction take note of signs a) 4x + 2y – 3xb) 3a – 4b – 6a + 9b c) -5d + 4a + 2 + 6d - 5a = (4 – 3)x = x + 2y= (3 - 6)a = -3a + 5b = -a + q + 2 d) 2a – 5b – 6c (- 5 + 6)d = 2a – 5b – 6c + 2y (- 4 + 9)b = (+ 4 - 5)a+ 2 If the number left in front of a letter is 1, it can be left out e.g. Simplify the following expressions:

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- Involves replacing variables with numbers and calculating the answer - Remember the BEDMAS rules e.g. If a = 7, b = 3, and c = 5, calculate the following a) a + b + cb) b - a c) abc = 7 + 3 + 5 = 15 = 3 - 7 = -4 = 7 × 3 × 5 = 105 - Or replacing numbers into formulas e.g. The area of a triangle is given by the formula A = ½b × h Find the area of: 12 m 5 mArea = ½ × 12 × 5 = 30 m 2 - Can substitute using mixed operations e.g. If a = 2, and b = 5, calculate the following a) 5b – 4ab) ab + 15= 5 × 5 – 4 × 2 = 25 - 8 = 2 × 5 + 15 = 10 + 15 = 17= 25

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- But always calculate brackets first! e.g. If a = 2, and b = 5, calculate the following a) 4(a + b)= 4(2 + 5) = 4 × 7 = (2 × 5 – 2) 4 = (10 – 2) 4 = 28 = 8 4 Because the top needs to be calculated first, brackets are implied b) 2b – a 4 = 2 - Can involve powers too which are done after brackets e.g. If a = 2, and b = 5, calculate the following a) a 2 b) (b – 2) 2 c) 4a 2 = 2 2 = 4 = (5 – 2) 2 = 3 2 = 4 × 2 2 = 4 × 4 d) 2b 2 – 3a 2 = 9 = 16 = 2 × 5 2 - 3 × 2 2 = 2 × 25 - 3 × 4 = 50 - 12 = 38

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- Remember: 3 × 3 × 3 × 3 =3434 - Variables (letters) that are multiplied by themselves are treated the same way e.g. Simplify these expressions that are written in full a) r × rb) p × p × p × p × p= r 2 = p 5 - Sometimes there may be two or more variables e.g. Simplify a) a × a × b × b × bb) d × e × e × d × f= a 2 b 3 = d 2 e 2 f Letters should still be written in alphabetical order! - With numbers and variables, multiply numbers and combine variables e.g. Simplify a) 4x × 6xb) 6b × 3b × 2 c) 2a × 5b × 3a × b = 4 × 6 × x × x = 24x 2 = 6 × 3 × 2 × b × b = 36b 2 = 2 × 5 × 3 × 1 × a × a × b × b = 30a 2 b 2 1

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- Does x 2 × x 3 = x × x × x × x × x ?YES - Therefore x 2 × x 3 = x 5 - How do you get 2 3 = 5 ?+ - When multiplying index (power) expressions with the same letter, ADD the powers. e.g. Simplify a) p 10 × p 2 b) a 3 × a 2 × a= p (10 + 2) = a (3 + 2 + 1) = p 12 1 No number = 1 i.e. p = 1p 1 = a 6 - Remember to multiply any numbers in front of the variables first e.g. Simplify a) 2x 3 × 3x 4 b) 2a 2 × 3a × 5a 4 = 2 × 3 = 2 × 3 × 5= 6 1 = 30 x (3 + 4) a (2 + 1 + 4) x7x7 a7a7

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YES- Does 6 = 1 ? 6 - Therefore x = 1 x - Does x 5 = x × x × x × x × x ? x 3 x × x × x YES= x × x × 1 × 1 × 1- Therefore x 5 = x 2 x 3 - When dividing index (power) expressions with the same letter, SUBTRACT the powers. e.g. Simplify a) p 5 ÷ p= p (5 - 1) = x (7 - 4) = p 4 = x 3 - Remember to divide any numbers in front of the variables first e.g. Simplify a) 12x 5 ÷ 6x 4 = 12 ÷ 6 = 2 - How do you get 5 3 = 2 ?- 1 b) x 7 x 4 If the power remaining is 1, it can be left out x (5 - 4) b) 5a 7 15a 2 ÷ 5 = 1 5 a (7 - 2) = 1 5 x a5a5 or a 5 5

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- 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 = 2x 2 1 = -3x × 2x- -3x × 5 = -6x 2 Don’t forget to watch for sign changes! + 6y+ 4y + 24- 14 + 3xy+ 15x

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- 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

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- Factorising is the reverse of expanding - To factorise:1) Look for a common factor to put outside the brackets 2) 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 - 15b) 30a + 20= 3( )2x2x- 5= 10( ) 3a 3a + 2 - Always look for the highest common factor You should always check your answer by expanding it e.g. Factorise a) 6x + 3b) 20b - 10= 3( )2x+ 1= 10( ) 2b 2b - 1 - Sometimes a ‘1’ will need to be left in the brackets - 6c

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e.g. Factorise a) cd - ceb) xyz + 2xy – 3yz= c( )d- e= y( ) xz + 2x - Letters can also be common factors e.g. Factorise a) 5a 2 – 7a 5 b) 4b 2 + 6b 3 = a 2 ( )5- 7a 3 = 2b 2 ( ) 2 + 3b - Powers greater than 1 can also be common factors - 3z c) 4ad – 8a= 4 ( )ad- 2

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- 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 division e.g. Solve 3x = 18 ÷3 x = 6 2) To undo addition we use subtraction e.g. Solve x + 2 = 6 -2 x = 4 3) To undo subtraction we use addition e.g. Solve x - 8 = 11 +8 x = 19 4) To undo division we use multiplication ×5 x = 30 e.g. Solve x = 6 5 ×5

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- Terms containing the variable (x) should be placed on one side (often left) e.g. Solve a) 5x = 3x + 6b) -6x = -2x + 12 -3x 2x = 6 ÷2÷2÷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+2x+2x -4x = 12 ÷-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 = 13b) -3x + 10 = 31 +5 6x = 18 ÷6÷6÷6÷6 x = 3 -10 -3x = 21 ÷-3 x = -7 Always look at the sign in front of the term/number to decide operation Don’t forget the integer rules!

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- Same rules apply for combined equations e.g. Solve a) 5x + 8 = 2x + 20b) 4x - 12 = -2x + 24 -2x 3x + 8 = 20 -8-8-8-8 3x = 12 +2x 6x - 12 = 24 ÷6÷6÷6 x = 6 ÷3÷3÷3 x = 4 +12 6x = 36 - Answers can also be negatives and/or fractions e.g. Solve a) 8x + 3 = -12x - 17b) 5x + 2 = 3x + 1 +12x 20x + 3 = -17 -3-3-3-3 20x = -20 -3x 2x + 2 = 1 ÷2÷2÷2 x = -1 2 ÷20 x = -1 -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

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- Expand any brackets first e.g. Solve a) 3(x + 1) = 6b) 2(3x – 1) = x + 8 -3-3-3-3 3x = 3 ÷3÷3÷3 x = 1 3x3x+ 3= 66x6x- 2= x + 8 -x 5x - 2 = 8 +2 5x = 10 ÷5÷5÷5 x = 2

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- Involves writing an equation and then solving e.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 n5- 4= n + 18 the same as if 18 were added to the number e.g. Write an equation for the following information and solve a) 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 Therefore width is 12 m xx x + 5 x + 5 + x + x + 5 + x = 58 4x + 10 = 58 4x = 48 ÷4÷4÷4÷4 x = 12

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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-4n 3n = 15 Therefore the number is 5 74 ÷3÷3÷3 n = 5

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