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ALGEBRA

Expressions and Equations. ALGEBRA. FINDing rules for linear patterns. - 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.

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ALGEBRA

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  1. Expressions and Equations ALGEBRA

  2. FINDing rules for linear patterns - 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. 3×1= 3 4×1= 4 + 3 + 4 3 = 1 - 2 4 = 5 + 1 + 3 + 4 + 3 + 4 3×4 – 2 4×4 + 1 + 3 + 4 Rule: s = 3×n - 2 Rule: d = 4×n + 1 1. Find the difference between terms and if the same multiply by n 2. Substitute to find constant 3. Check if rule works

  3. Note: × and ÷ are not often used in Algebra Creating expressions • 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 Let n = a number • a) A number with 12 added to it n + 12 • b) A number with 9 subtracted from it n - 9 • c) A number multiplied by 2 n × 2 Best written as 2n • d) A number divided by 6 n ÷ 6 Best written as n 6 As long as you explain what a symbol represents, any symbol can be used

  4. e.g. John has x dollars. How much will he have if: • a) He spends $35 x - 35 50 - 35 = $15 x + 28 50 + 28 = $78 • b) He is given $28 • c) He doubles his money 2 x 2 x 50 = $100 • d) He spends half x 50 = $25 2 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:

  5. Simplifying expressions by multiplying • - ALL terms can be multiplied • Rules: 1) Multiply all numbers in the expression • 2) Place letters in alphabetical order behind product No number = 1 i.e. -p = -1p • e.g. Simplify: • a) 4c × 2d = 4 × 2 × c × d • b) –p × -2q × -6r = -1 × -2 × -6 ×p × q× r = 8cd = -12pqr 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 = r2 • b) p × p × p × p × p = p5 • - Sometimes there may be two or more variables • e.g. Simplify • a) a × a × b × b × b = a2b3 • b) d × e × e × d × f = d2e2f Letters should still be written in alphabetical order!

  6. Four Power rules • 1. 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. No number = 1 i.e. p = 1p1 • e.g. Simplify • a) p10 × p2 = p(10 + 2) • b) a3 × a2 × a = a(3+ 2 + 1) 1 = p12 = a6 • - Remember to multiply any numbers in front of the variables first • e.g. Simplify • a) 2x3 × 3x4 = 2 × 3 x(3+ 4) • b) 2a2 × 3a × 5a4 1 = 6 x7 = 2 × 3 × 5 a(2 + 1 + 4) = 30 a7

  7. 2. Division - Does 6 = 1 ? 6 • YES - 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 • - How do you get 5 3 = 2 ? • - • - When dividing index (power) expressions with the same letter, SUBTRACT the powers. • e.g. Simplify • a) p5 ÷ p = p(5 - 1) b) x7 x4 = x(7 - 4) 1 = p4 = x3 • - Remember to divide any numbers in front of the variables first • e.g. Simplify • a) 12x5 ÷ 6x4 = 12 ÷ 6 x(5- 4) b) 5a7 15a2 ÷ 5 ÷ 5 = 1 5 a(7- 2) = 2 x = 1 5 a5 or a5 5 If the power remaining is 1, it can be left out of the answer

  8. 3. Powers of powers • - Does (x2)3 = x2 × x2 × x2 ? • YES • - Does x2 × x2 × x2 = x6 ? • YES • - Therefore (x2)3 = x6 • - How do you get 2 3 = 6 ? • × • - When taking a power of an index expression to a power, MULTIPLY the powers • e.g. Simplify • a) (c4)6 = c(4 × 6) • b) (a3)3 = a(3× 3) = c24 = a9 • - If there is a number in front, it must be raised to the power, not multiplied • e.g. Simplify • a) (3d2)3 = 33 × d(2 × 3) • b) (2a3)4 = 24 × a(3 × 4) = 27 d6 = 16 a12 • - If there is more than one term in the brackets, raise all to the power • e.g. Simplify • b) (4b2c5)2 = 42 b(2 × 2) c(5 × 2) • a) (x3y z4)3 = x(3 × 3) y(1 × 3) z(4 × 3) 1 = x9 y3 z12 = 16 b4 c10

  9. 4. Powers of zero • - Any base to the power of zero has a value of 1 • e.g. x0 = 1 Square roots • - Simply halve the power (as √x is the same as x½) • e.g. Simplify • a) • b) = 10 x4 = 8 x3 LIKE and UNLIKETERms • - LIKE terms are those with exactly the same letter, or combination of letters and powers • LIKE terms: • 2x, 3x, 31x • UNLIKE terms: • 2x, 3 • 4ab, 7ab • 5x, 6x2 • 2ab,2ac • e.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.

  10. 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 letters • e.g. Simplify these expanded expressions into compact form: • a) a + a + a 1 1 1 = (1 + 1 + 1)a • b) 5x + 6x + 2x = (5 + 6 + 2)x = 3a = 13x • c) 3p + 7q + 2p + 5q = (3 + 2)p (+7 + 5)q • d) 4a + 3a2 + 7a + a2 1 = 5p+12q = (4 + 7)a (+3 + 1)a2 = 11a+4a2 • - For expressions involving both addition and subtraction take note of signs • e.g. Simplify the following expressions: • a) 4x + 2y – 3x = (4 – 3)x + 2y • b) 3a – 4b – 6a + 9b = x+ 2y = (3 - 6)a (- 4 + 9)b = -3a+ 5b • c) 3x2 - 9x + 6x2 + 8x - 5 = (3 + 6)x2 (- 9 + 8)x - 5 = 9x2- x - 5 If the number left in front of a letter is 1, it can be left out • d) 4ab2 +2a2b – 5ab2 + 3ab = -ab2 + 2a2b + 3ab

  11. Expanding expressions • - Does 6 × (3 + 5) = 6 × 3 + 6 × 5 ? • YES • 6 × 8 = 18 + 30 • 48 = 48 • - 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) = 6 × x + 6 × y • b) -4(x – y) = -4 × x - -4 × y = 6x + 6y = -4x + 4y • c) -4(x – 6) = -4 × x - -4 × 6 • d) 7(3x – 2) = 7 × 3x - 7 × 2 = -4x + 24 = 21x - 14 • e) x(2x + 3y) = x × 2x 1 1 + x × 3y • f) -3x(2x – 5) = -3x × 2x - -3x × 5 = 2x2 + 3xy = -6x2 + 15x Don’t forget to watch for sign changes!

  12. - 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) = 2 × 4x + 2 × y + 8 × 3x - 8 × 2y = 8x + 2y + 24x - 16y = 32x - 14y • b) -3(2a – 3b) – 4(5a + b) = -3 × 2a - -3 × 3b - 4 × 5a + -4 × 1b = -6a + 9b - 20a - 4b = -26a + 5b

  13. 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 = (2 × 5 – 2) 4 = (10 – 2) 4 Because the top needs to be calculated first, brackets are implied = 8 4 = 2

  14. 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: 3a 6a ÷ 3 ÷ 3 1 1 = 1 2 a(1 – 1) b) 18x 6y ÷ 6 ÷ 6 = 3 1 x y = 1 2 = 3x y c) 4x3 6x2 ÷ 2 ÷ 2 = 2 3 x(3 – 2) = 2 3 x

  15. 2. Multiplying Fractions • - Multiply top and bottom terms separately then simplify. • e.g. Simplify: a) b × b2 2 5 = b × b2 b) y2 × 4 3 y = y2 × 4 2 × 5 3 × y = b3 10 = 4y2 3y = 4y 3 • 3. Dividing Fractions • - Multiply the first fraction by the reciprocal of the second, then simplify Note: b is the reciprocal of 2 2 b • e.g. Simplify: a) 2a ÷ a2 5 3 = 2a 5 × 3 a2 = 2a × 3 5 × a2 = 6a 5a2 = 6 5a or 6a-1 5

  16. 4. Adding/Subtracting Fractions • a) With the same denominator: • - Add/subtract the numerators and leave the denominator unchanged. Simplify if possible. • e.g. Simplify: a) 3x + 3x 10 10 = 3x + 3x 10 b) 6a - b 5 5 = 6a - b 5 = 6x 10 ÷ 2 ÷ 2 = 3x 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: b) 2x – 5x 3 4 = 3×4 4×2x - 3×5x a) a + 2a 2 3 = 2×3 3×a + 2×2a = 8x – 15x 12 = 3a + 4a 6 = -7x 12 = 7a 6

  17. factorising expressions • - 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 You should always check your answer by expanding it • e.g. Factorise • a) 2x + 2y = 2( ) x + y • b) 2a + 4b – 6c = 2( ) a + 2b - 3c • - Always look for the highest common factor • e.g. Factorise • a) 6x - 15 = 3( ) 2x - 5 • b) 30a + 20 = 10( ) 3a + 2 • - Sometimes a ‘1’ will need to be left in the brackets • e.g. Factorise • a) 6x + 3 = 3( ) 2x + 1 • b) 20b - 10 = 10( ) 2b - 1

  18. - Letters can also be common factors • e.g. Factorise • a) cd - ce = c( ) d - e • b) xyz + 2xy – 3yz = y( ) xz + 2x - 3z • c) 4ad – 8a = 4 ( ) a d - 2 • - Powers greater than 1 can also be common factors • e.g. Factorise • a) 5a2 – 7a5 = a2( ) 5 - 7a3 • b) 4b2 + 6b3 = 2b2( ) 2 + 3b

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

  20. - 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 +2x 2x = 6 -4x = 12 Don’t forget the integer rules! ÷2 ÷2 ÷-4 ÷-4 x = 3 x = -3 Always line up equals signs and each line should contain the variable and one equals sign You should always check your answer by substituting into original equation Always look at the sign in front of the term/number to decide operation • YES - Does 5×3 = 3×3 + 6 ? - Does -6×-3 = -2×-3 + 12 ? • YES 15 = 9 + 6 18 = 6 + 12 • - 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 -10 -10 6x = 18 -3x = 21 ÷6 ÷6 ÷-3 ÷-3 x = 3 x = -7

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

  22. - Expand any brackets first • e.g. Solve a) 3(x + 1) = 6 b) 2(3x – 1) = x + 8 3x + 3 = 6 6x - 2 = x + 8 -3 -3 -x -x 3x = 3 5x - 2= 8 ÷3 ÷3 +2 +2 x = 1 5x = 10 ÷5 ÷5 x = 2 • - For fractions, cross multiply, then solve • e.g. Solve a) x = 9 4 2 b) 3x - 1 = x + 3 5 2 2x = 36 2(3x - 1) = 5(x + 3) ÷2 ÷2 x = 18 6x - 2 = 5x + 15 -5x -5x x - 2= 15 +2 +2 x = 17

  23. - For two or more fractions, find a common denominator, multiply it by each term, then solve e.g. Solve 4x - 2x = 10 5 3 ×15 ×15 ×15 5 × 3 = 15 60x 5 - 30x 3 = 150 Simplify terms by dividing numerator by denominator 12x - 10x = 150 2x = 150 ÷2 ÷2 x = 75

  24. Writing EQUATIONS and solving • - 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. Let n = a number 3 n + 12 = 36 • a) I think of a number, multiply it by 5 and then subtract 4. The result is the same as if 18 were added to the number Let n = a number 5 n - 4 = n + 18 • 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 x + 5 x + 5 + x + x + 5 + x = 58 4x + 10 = 58 Draw a diagram -10 -10 x x 4x = 48 Let x = width ÷4 ÷4 x + 5 x = 12 Therefore width is 12 m

  25. 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 7 n = n 4 + 15 -4n -4n 3n = 15 ÷3 ÷3 n = 5 Therefore the number is 5 Solving inequations • - 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. Solve a) 3x + 8 >24 b) -2x -5 ≤ 13 -8 -8 +5 +5 3x >16 -2x ≤ 18 As answer not a whole number, leave as a fraction ÷3 ÷3 ÷-2 ÷-2 Sign reverses as dividing by a negative x >16 3 x ≥ -9

  26. Changing the subject • - Involves rearranging the formula in order to isolate the new ‘subject’ • - Same rules as for solving are used Remember: When rearranging or changing the subject you are NOT finding a numerical answer • e.g. • a) Make x the subject of y = 6x - 2 +2 +2 y + 2 = 6x ÷6 ÷6 All terms on the left must be divided by 6 y + 2 = x 6 • b) Make R the subject of IR = V Treat letters the same as numbers! ÷I ÷I R =V I • c) Make x the subject of y = 2x2 ÷2 ÷2 y = x2 2 Taking the square root undoes squaring

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