Simplifying Algebraic Expressions Topic 2.1.1

Simplifying Algebraic Expressions 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(2x – 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

Simplifying Algebraic Expressions 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: 3x + 2 – 7 – x can be written as: 2x – 5

Simplifying Algebraic Expressions 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 4x2 or 5x, or just a number. For example, the algebraic expression 4x2 – 5x + 7 – 2x2 + 2x – 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. 8x2 + 2xy – 6y + 7y2 – 2 Invisible + sign. This minus sign belongs to the 6y term.

Simplifying Algebraic Expressions Topic 2.1.1 Simplifying Algebraic Expressions Guided Practice Write the number of terms in the expressions in Exercises 1–4. 1. 3x2 + 4x – 2 2. 8x4 + 7x3 + 2x2 – 8 3. 3x2 4. 8 + 7xy – 2xy9 + 4x7 + 3y2 + 55y3 5. Which variable is multiplied by –4 in the algebraic expression 4x2 – 4y + 8 + 4xy? 6. Counting from left to right, which term is the fourth term in the algebraic expression 8x2 + 2xy – 6y + 9xy3 – 4? 3 4 1 6 y 9xy3 Solution follows…

Simplifying Algebraic Expressions 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 4x2 and –2x2 are like terms because they have the same variable, x, with the same exponent, 2. –5x and 2x are like terms, and 7 and –3 are like terms too. Like terms can be combined using the distributive, commutative, and associative properties.

Simplifying Algebraic Expressions Topic 2.1.1 Simplifying Algebraic Expressions Example 1 Simplify 4x2 – 2x2. Solution 4x2 – 2x2 = (4 – 2)x2 = 2x2 Solution follows…

Simplifying Algebraic Expressions 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.

Simplifying Algebraic Expressions Topic 2.1.1 Simplifying Algebraic Expressions Example 2 Simplify the following: a) (4x – 5) – 2x b) 5x2 – 3x + 7x2 – 4x + 9 Solution a) (4x – 5) – 2x = 4x + (–5 – 2x) Associative property of addition = 4x + (–2x – 5) Commutative property of addition = (4x + –2x) – 5 Associative property of addition = [(4 – 2)x] – 5 Distributive property = 2x – 5 Example continues… Solution follows…

Simplifying Algebraic Expressions Topic 2.1.1 Simplifying Algebraic Expressions Example 2 Simplify the following: a) (4x – 5) – 2x b) 5x2 – 3x + 7x2 – 4x + 9 Solution b) 5x2 – 3x + 7x2 – 4x + 9 = 5x2 + 7x2 – 3x – 4x + 9 = (5x2 + 7x2) + (–3x – 4x) + 9 = (5 + 7)x2 + (–3 – 4)x + 9 = 12x2 + (–7)x + 9 = 12x2 – 7x + 9

Simplifying Algebraic Expressions Topic 2.1.1 Simplifying Algebraic Expressions Guided Practice Simplify the following expressions in Exercises 7–14. 7. 7a + 12a – 4 8. 7a + 3b – 5a 9. 9x + (20 – 5x) 10. 5 – 10x – 2x – 7 11. 7x2 + 7 + 20x2 + 3x 12. 3 – 8x2 + 4x2 + 6x2 – 10 13. 5a – 4 × 3a + 7 × 2 – 3 × 6 14. 3 × 4a – 2 × 5a2 + 2 × 2a2 19a – 4 2a + 3b 4x + 20 –12x – 2 27x2 + 3x + 7 2x2 – 7 –7a – 4 –6a2 + 12a Solution follows…

Simplifying Algebraic Expressions Topic 2.1.1 Simplifying Algebraic Expressions Guided Practice In Exercises 15–18, simplify the expressions and determine the number of terms in each simplified form. 15. (4a – 9) + (2a – 18) 16. 15n + 3n + 8 – 2 – 6 17. 6a2 + 3a – 9a2 + 2a + 7 + 6 18. 6a + 3 × 7b – 2 × 5c + 7 – 9 + 2 × 4c 6a – 27; 2 terms 18n; 1 term –3a2 + 5a + 13; 3 terms 6a + 21b – 2c – 2; 4 terms Solution follows…

Simplifying Algebraic Expressions Topic 2.1.1 Simplifying Algebraic Expressions Independent Practice In Exercises 1–5, determine the number of terms in each algebraic expression: 1. 7b + 14a – 4 2. 2a 3. (27x2 + 4x) – 13 4. 5 +10x + 20x2 + 3a 5. 2x + 4xy + 4x2 – (10 + 12y + 19y2) 3 1 3 4 6 Solution follows…

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

Simplifying Algebraic Expressions Topic 2.1.1 Simplifying Algebraic Expressions Independent Practice In Exercises 10–13, simplify each algebraic expression: 10. 3x + y – x 11. a – b + b – a 12. (17.8n + 13.08q) – 3.8q – 9.9n 13. 0.4x2 + x – x2 – 0.14 – 1 2 8 3 10 6 12 5 1 2 2 x + y 3 8 – a + b 1 2 7.9n + 9.28q 3 8 –0.1x2 + x – 0.765 Solution follows…

Simplifying Algebraic Expressions Topic 2.1.1 Simplifying Algebraic Expressions Independent Practice 14. Which expression below simplifies to 2x + 1? i. x + 4x + 4 – 3 – 3x + x ii. 7 + 5x – 6 + 5 – 4x + x – 5 ii) simplifies to 2x + 1 15. Which expression below simplifies to 3x? i. 5 + 4x2 – 3x – 2 + 4x2 – 8x2 – 3 ii. –2x2 + 6 + 4x2 – 3 + 3x – 2x2 – 3 ii) simplifies to 3x Solution follows…

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

Simplifying Algebraic Expressions Topic 2.1.1 Simplifying Algebraic Expressions Independent Practice 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 (2x + 4) pieces of candy, Leo has (8 – 2x) 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. 3b + 2c + 20 20 Solution follows…

Simplifying Algebraic Expressions 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.