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Worksheet 2.2 Solving Equations in One Variable Section 1 Simple Examples You are on your way to Brisbane from Sydney, and you know that the trip is 1100 km. You pass a sign that says that Brisbane is now 200 km away. How far have you travelled? We can see the answer quickly, but we shall write the information down in the form of a mathematical expression. Let x be the distance travelled so far. Then x + 200 = 1100 This is the information that we are given. To find the answer we are to solve an equation in one variable, x. x + 200 = 1100 x + 200 − 200 = 1100 − 200 subtract 200 from both sides x = 900 The middle step emphasizes what we do in the calculation. We wanted x by itself on one side of the equation, so we subtracted 200 from each side of the equation. An equation is like a balancing beam in the sense that if you do anything to one side of the equation, you must also do it to the other. Example 1 : If x + 2 = 3, what is x? Clearly, x = 1, but again we follow through the calculation. Example 2 : If x − 5 = −12, what is x? x + 2 = 3 x+ 2− 2 = 3 − 2 x = 1 x− 5+ 5 = −12 + 5 adding 5 to both sides x = −7 Example 3 : Solve for x in −8 = −10 + x. −8 = −10 + x −8 + 10 = −10 + x + 10 x = 2 Page 10
In each of the above calculations, we worked on the equation by trying to get x on one side and all the rest of the expression on the other side. This is a basic method for solving equations in one variable. To simplify the expression involving x we simply undid whatever was done to x by applying the opposite arithmetic operation to leave us with just an x on one side. Most importantly, we need to remember that whatever we do to one side of the equation, we must do to the other. For example, if we subtract 5 from one side of the equation, we must subtract 5 from the other. Exercises: 1. Solve the following equations: (a) x + 9 = 20 (b) x − 8 = 10 (c) x + 1.6 = 2.4 (d) x − 3 1 = 1 4 2 (e) y − 2 = −8 (f) m + 6 = −4 (g) m − 7.1 = −8.4 (h) t − 2.4 = −1 (i) x + 1 1 2 = 2 (j) y + 80 = 120 Section 2 Equations Involving Multiplication In a similar way to the above we can deal with solving equations such as 5x = 2 The arithmetic operation is now multiplication, whose inverse is division. The inverse of multiplying by 5 is dividing by 5. 5x = 2 becomes 5 x 5 = 2 5 dividing both sides by 5 1x = 2 x = 5 2 5 Page 11
Page 1 and 2: Table of Contents Module 1 1.1 Orde
Page 3 and 4: Worksheet 2.1 Factors of Algebraic
Page 5 and 6: By factorizing we can make a slight
Page 7 and 8: Example 3 : 4a + 2ab = 2a 2a(2 + b)
Page 9: 1. What are the factors of 18? 2. W
Page 13 and 14: Solution 1 Solution 2 Putting in J
Page 15 and 16: Exercises: 1. Solve the following e
Page 17 and 18: Worksheet 2.3 Algebraic Fractions S
Page 19 and 20: y a fraction is the same operation
Page 21 and 22: Section 3 Solving Equations Sometim
Page 23 and 24: 2. Solve for x in terms of y: (a) 3
Page 25 and 26: Worksheet 2.4 Introduction to Inequ
Page 27 and 28: Example 7 : The number of matches i
Page 29 and 30: Example 6 : Exercises: Example 7 :
Page 31 and 32: Example 3 : this is the same as Exa
Page 33 and 34: Exercises for Worksheet 2.4 1. (a)
Page 35 and 36: Note: After expansion of this expre
Page 37 and 38: Example 2 : Example 3 : √ √ 7 +
Page 39 and 40: Example 4 : Example 5 : Example 6 :
Page 41 and 42: 1. (a) √ 2 × √ 2 (b) ( √ 5)
Page 43 and 44: Exercises: 1. Factorize the followi
Page 45 and 46: Exercises: 1. Expand the following,
Page 47 and 48: Section 4 Factorizing Quadratics Be
Page 49 and 50: Example Factorize 6x 2 − x − 12
Page 51 and 52: the quadratic formula. Note that a
Page 53 and 54: Example 3 : Solve (x + 3) 2 = x + 5
Page 55 and 56: Section 7 Multiplication and Divisi
Page 57 and 58: Worksheet 2.7 Logarithms and Expone
Page 59 and 60: Note that for all of the above prop
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exactly in decimal form, but it is
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1. Evaluate (a) log 10 1000 (b) log
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Example 4 : If CD and AB are parall
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The triangle ABC in the above examp
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Section 3 Pythagoras’ Theorem All
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Section 4 Introductory Trigonometry
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Exercises: 1. Find the following ra
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Worksheet 2.9 Introduction to the C
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Exercises: 1. Complete the table of
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Example 1 : y = 2x + 3. The gradien
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Example 8 : The lines y = 10x + 3 y
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which doesn’t make sense as we ca
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2. (a) Plot the points X(−1, 2) a
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Worksheet 2.1 Section 1 1. (a) 6 (b
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2. (a) -1 (b) 46,47 (c) 22,24 Works
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Exercises 2.4 1. (a) i. ii. iii. iv
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2. (a) 13 √ 3 (b) −12 √ 5 (c)
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2. (a) 5x2 +13x+9 (x+2)(x+3) (b) 2(
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Exercises 2.7 1. (a) 3 (b) 0 (c) 3
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2. (a) Section 2 1. (a) & (d) Secti
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