Grade 8 Simplifying Expressions and Solving Equations Cambridge [PPT]
- 1. Simplifying expressions and solving equations Grade 8
- 2. Topic(s) • Collecting like terms • Expanding brackets • Constructing and solving equations nomadic-math.blogspot.com
- 3. Collecting like terms • There are some rules you can follow when you write an expression in algebra : - Write products without the multiplication sign, so write 2 × 𝑛 as 2𝑛 - Write the number before the letter, so write 2𝑥 not 𝑥2 - Generally, write terms with letters before number terms, so write 3𝑦 + 4 rather than 4 + 3𝑦 - Generally, write terms in alphabetical order, so write 4𝑎 + 5𝑏 not 5𝑏 + 4𝑎 - When a term has more than one letter, write them in alphabetical order, so write 6𝑐𝑑 rather than 6𝑑𝑐 - Write negative terms after positive terms, so write 5 − 4𝑧 not −4𝑧 + 5 (unless all the terms are negative, in which case follow the order rules, so write − 3𝑥 − 8𝑦, not −8𝑦 − 3𝑥) Algebraic expression nomadic-math.blogspot.com
- 4. • Like terms are terms that contain the same letter variables which are raised to the exact same powers. • You can also simplify expressions by collecting like terms 3𝑎 + 𝑏 + 2𝑎 + 6𝑏 2𝑥𝑦2 + 𝑥 − 𝑥𝑦2 + 𝑥2 𝑐2 + 𝑑3 − 2𝑐2 example Simplify these expressions a. 4 + 2𝑥 + 5𝑥 b. 2𝑎𝑏 + 𝑎𝑏 − 5𝑏𝑎 c. 2𝑦 + 6𝑦2 − 3𝑦2 − 10𝑦 a. 2𝑥 + 5𝑥 + 4 = 7𝑥 + 4 2𝑥 and 5𝑥 are like terms b. 2𝑎𝑏 + 𝑎𝑏 − 5𝑎𝑏 = −2𝑎𝑏 5𝑏𝑎 is the same as 5𝑎𝑏 c. 6𝑦2 − 3𝑦2 + 2𝑦 − 10𝑦 = 3𝑦2 − 4𝑎𝑏 6𝑦2 and −3𝑦2 like terms, also 2𝑦 and −10𝑦 nomadic-math.blogspot.com
- 5. exercise 1. Use the guidelines opposite to rewrite these expressions a. 8 × 𝑛 b. 4 + 3 × 𝑣 c. 6 × 𝑚 × 𝑛 d. 𝑝 × 2 + 𝑞7 e. −3𝑦𝑥 − 8𝑏𝑎 2. Simplify each expressions a. 6𝑥 + 5𝑥 + 9𝑥 b. 8𝑐 − 4𝑑 − 2𝑐 + 𝑑 c. 4𝑥2 + 5𝑥2 + 8𝑥 − 5𝑥 d. 2𝑎𝑏 + 7𝑎𝑏 − 7𝑏𝑎 e. 11𝑦2 − 3𝑦 − 5𝑦2 f. 𝑎2 + 2𝑎 + 2 − 𝑎2 + 5𝑎 nomadic-math.blogspot.com
- 6. Expanding brackets • To expand brackets, multiply each term inside the brackets by the term outside the brackets example a. Expand 3 𝑏 + 6 b. Expand and simplify 4 2𝑥 + 3𝑥2 − 𝑥 2 + 𝑥 a. 3 𝑏 + 6 = 3𝑏 + 18 b. 4 2𝑥 + 3𝑥2 − 𝑥 2 + 𝑥 = 8𝑥 + 12𝑥2 − 2𝑥 + 𝑥2 = 8𝑥 + 12𝑥2 − 2𝑥 − 𝑥2 = 12𝑥2 − 𝑥2 + 8𝑥 − 2𝑥 = 11𝑥2 + 6𝑥 nomadic-math.blogspot.com
- 7. exercise 1. Expand each expression a. 3(𝑥 + 7) b. 12(3𝑎 − 4) c. 2(𝑎𝑏 + 4𝑐) d. 𝑥(3𝑦 + 4) e. 𝑔(3ℎ + 6𝑔) f. 2𝑓(2𝑓 + 𝑔 − 2) 2. Expand and simplify each expression a. 2 𝑥 + 3 + 2(𝑥 + 2) b. 5 5 + 4𝑣 − 4(3𝑣 + 7) c. 8 𝑧 + 3 + 5(4 + 3𝑧) d. 𝑥 𝑥 + 2 + 𝑥(𝑥 + 4) e. 𝑢 2𝑢 − 6 − 𝑢 𝑢 + 3 f. 𝑦 𝑦 + 2𝑥 + 4(𝑥 − 2) nomadic-math.blogspot.com
- 8. Constructing and solving equations • When you are given a problem to solve, you may need to construct, or write, an equation to help you solve the problem example The diagram shows a rectangle. Work out the values of 𝑥 and 𝑦 3(𝑥 + 3) cm 3𝑦 + 8 cm5𝑦 − 4 cm 24 3 𝑥 + 3 = 24 3𝑥 + 9 = 24 3𝑥 + 9 − 9 = 24 − 9 3𝑥 = 15 𝒙 = 𝟓 5𝑦 − 4 = 3𝑦 + 8 5𝑦 − 3𝑦 = 8 + 4 2𝑦 = 12 𝒚 = 𝟔 The two lengths must be equal to find 𝑥 The two widths must be equal to find 𝑦 nomadic-math.blogspot.com
- 9. exercise 1. Work out the value 𝑥 and 𝑦 in each of these diagrams. All measurements are centimetres 3𝑥 + 1 2𝑦 + 154𝑦 + 5 2(𝑥 + 5) 5𝑥 − 3 8𝑦 − 43𝑦 + 16 3𝑥 + 11 3𝑥 165𝑦 + 1 18 6(𝑥 + 1) 202(𝑦 + 3) 72 a. b. c. d. nomadic-math.blogspot.com
- 10. 2. Work out the value of 𝑥 or 𝑦 in each of these shapes. All measurements are centimetres 27 3(𝑥 + 5) 8𝑦 − 5 3(𝑦 + 5) a. b. nomadic-math.blogspot.com
- 11. Source : Cambridge Checkpoint Mathematics 8 nomadic-math.blogspot.com