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1 Topic 2.1.1 Simplifying Algebraic Expressions Simplifying Algebraic Expressions

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2 Topic 2.1.1 Simplifying Algebraic Expressions California Standard: 4.0 Students simplify expressions before solving linear equations and inequalities in one variable, such as 3(2 x – 5) + 4( x – 2) = 12. What it means for you: You’ll combine like terms to simplify expressions. Key words: algebraic expression simplify like terms

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3 Topic 2.1.1 Simplifying Algebraic Expressions Unless you’re told otherwise, you always need to give algebraic solutions in the simplest form possible. For example: 3 x + 2 – 7 – x can be written as: 2 x – 5

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4 Topic 2.1.1 Simplifying Algebraic Expressions Algebraic Expressions Contain Variables Algebraic expressions are made up of terms. A term can be a product of numbers and variables, like 4 x 2 or 5 x, or just a number. For example, the algebraic expression 4 x 2 – 5 x + 7 – 2 x 2 + 2 x – 3 has six terms. The terms are separated by plus and minus signs. Each sign “belongs” to the term that it’s in front of. 8 x 2 + 2 xy – 6 y + 7 y 2 – 2 Invisible + sign. This minus sign belongs to the 6 y term.

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5 Topic 2.1.1 Simplifying Algebraic Expressions Guided Practice Solution follows… Write the number of terms in the expressions in Exercises 1–4. 1.3 x 2 + 4 x – 2 2.8 x 4 + 7 x 3 + 2 x 2 – 8 3.3 x 2 4. 8 + 7 xy – 2 xy 9 + 4 x 7 + 3 y 2 + 55 y 3 5. Which variable is multiplied by –4 in the algebraic expression 4 x 2 – 4 y + 8 + 4 xy ? 6.Counting from left to right, which term is the fourth term in the algebraic expression 8 x 2 + 2 xy – 6 y + 9 xy 3 – 4? 3 4 1 6 y 9 xy 3

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6 Topic 2.1.1 Simplifying Algebraic Expressions Like Terms Can Be Combined Like terms are terms with identical variables that have identical exponents. The terms 4 x 2 and –2 x 2 are like terms because they have the same variable, x, with the same exponent, 2. –5 x and 2 x are like terms, and 7 and –3 are like terms too. Like terms can be combined using the distributive, commutative, and associative properties.

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7 Topic 2.1.1 Simplifying Algebraic Expressions Example 1 Solution follows… Simplify 4 x 2 – 2 x 2. Solution 4 x 2 – 2 x 2 = (4 – 2) x 2 = 2 x 2

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8 Topic 2.1.1 Simplifying Algebraic Expressions Combine Like Terms to Simplify Expressions To simplify an algebraic expression, use number properties to first group and then combine like terms.

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9 Topic 2.1.1 Simplifying Algebraic Expressions Example 2 Solution follows… Simplify the following: Solution a) a) (4 x – 5) – 2 x (4 x – 5) – 2 x 4 x + (–5 – 2 x )= = = = = 4 x + (–2 x – 5) (4 x + –2 x ) – 5 [(4 – 2) x ] – 5 2 x – 5 Associative property of addition Commutative property of addition Associative property of addition Distributive property Example continues… b) 5 x 2 – 3 x + 7 x 2 – 4 x + 9

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10 Topic 2.1.1 Simplifying Algebraic Expressions Example 2 Simplify the following: Solution b) b) 5 x 2 – 3 x + 7 x 2 – 4 x + 9 5 x 2 – 3 x + 7 x 2 – 4 x + 9 5 x 2 + 7 x 2 – 3 x – 4 x + 9= = = = = (5 x 2 + 7 x 2 ) + (–3 x – 4 x ) + 9 (5 + 7) x 2 + (–3 – 4) x + 9 12 x 2 + (–7) x + 9 12 x 2 – 7 x + 9 a) (4 x – 5) – 2 x

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11 Simplify the following expressions in Exercises 7–14. 7. 7 a + 12 a – 48. 7 a + 3 b – 5 a 9. 9 x + (20 – 5 x )10. 5 – 10 x – 2 x – 7 11. 7 x 2 + 7 + 20 x 2 + 3 x 12. 3 – 8 x 2 + 4 x 2 + 6 x 2 – 10 13. 5 a – 4 × 3 a + 7 × 2 – 3 × 6 14. 3 × 4 a – 2 × 5 a 2 + 2 × 2 a 2 Topic 2.1.1 Simplifying Algebraic Expressions Guided Practice Solution follows… 19 a – 42 a + 3 b 4 x + 20–12 x – 2 27 x 2 + 3 x + 72 x 2 – 7 –7 a – 4 –6 a 2 + 12 a

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12 In Exercises 15–18, simplify the expressions and determine the number of terms in each simplified form. 15. (4 a – 9) + (2 a – 18) 16. 15 n + 3 n + 8 – 2 – 6 17. 6 a 2 + 3 a – 9 a 2 + 2 a + 7 + 6 18. 6 a + 3 × 7 b – 2 × 5 c + 7 – 9 + 2 × 4 c Topic 2.1.1 Simplifying Algebraic Expressions Guided Practice Solution follows… 6 a – 27; 2 terms 18 n ; 1 term –3 a 2 + 5 a + 13; 3 terms 6 a + 21 b – 2 c – 2; 4 terms

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13 Topic 2.1.1 Simplifying Algebraic Expressions Independent Practice Solution follows… In Exercises 1–5, determine the number of terms in each algebraic expression: 1. 7 b + 14 a – 42. 2 a 3. (27 x 2 + 4 x ) – 134. 5 +10 x + 20 x 2 + 3 a 5. 2 x + 4 xy + 4 x 2 – (10 + 12 y + 19 y 2 ) 31 34 6

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14 Topic 2.1.1 Simplifying Algebraic Expressions Independent Practice Solution follows… In Exercises 6–9, simplify each algebraic expression: 6. 12 m – 7 + 3 c – 7 m – 8 c 7. 4 a + 3 b + 11 a – 8 b 8. 2 × 6 x – 3 × 5 x – 3 x × 4 + 5 × 2 x + 12 x 9. 5 m × 3 + 2 × 7 m – 4 m × 4 + 7 × 2 m – 17 m 5 m – 5 c – 7 15 a – 5 b 7x7x 10 m

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15 In Exercises 10–13, simplify each algebraic expression: 10. 3 x + y – x 11. a – b + b – a 12. (17.8 n + 13.08 q ) – 3.8 q – 9.9 n 13. 0.4 x 2 + x – x 2 – 0.14 – 1 2 1 2 1 8 3 10 8 6 12 3 8 1 2 5 8 Topic 2.1.1 Simplifying Algebraic Expressions Independent Practice Solution follows… 7.9 n + 9.28 q 1 2 2 x + y 1 2 3 8 – a + b 1 2 3 8 –0.1 x 2 + x – 0.765

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16 Topic 2.1.1 Simplifying Algebraic Expressions Independent Practice Solution follows… 14. Which expression below simplifies to 2 x + 1? i. x + 4 x + 4 – 3 – 3 x + x ii. 7 + 5 x – 6 + 5 – 4 x + x – 5 15. Which expression below simplifies to 3 x ? i. 5 + 4 x 2 – 3 x – 2 + 4 x 2 – 8 x 2 – 3 ii. –2 x 2 + 6 + 4 x 2 – 3 + 3 x – 2 x 2 – 3 ii) simplifies to 2 x + 1 ii) simplifies to 3 x

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17 Topic 2.1.1 Simplifying Algebraic Expressions Independent Practice Solution follows… In Exercises 16–17, find a simplified expression for the perimeter of the figure. 16. 17. 12 x + 2 4 a + 4 x + 10 4 x + 1 2x2x 3 2x2x 7 2x2x 2a2a 2a2a

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18 18. Juan bought 3 baseball cards for b dollars each and 2 baseball cards for c dollars each. He has bought 4 comic books for $5.00 each. Write and simplify an algebraic expression showing the total money Juan spent on baseball cards and comic books. 19. Three friends Tom, Leo, and Maria have several pieces of candy to eat. Tom has (2 x + 4) pieces of candy, Leo has (8 – 2 x ) pieces of candy, and Maria has 8 pieces of candy. Write and simplify an algebraic expression showing the total number of pieces of candy the three friends have to eat. Topic 2.1.1 Simplifying Algebraic Expressions Independent Practice Solution follows… 3 b + 2 c + 20 20

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19 Topic 2.1.1 Simplifying Algebraic Expressions Round Up You’ve combined like terms before, in earlier grades — so this Topic should feel like good practice. It’s always important to give your final answers to algebraic problems in the simplest form.

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