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Equations and Inequalities Copyright © Cengage Learning. All rights reserved. 2 Section 2.3 Simplifying Expressions to Solve Linear Equations in One Variable Copyright © Cengage Learning. All rights reserved. Objectives 1. Simplify an expression using the order of 1 operations and combining like terms. 22. Solve a linear equation in one variable requiring simplifying one or both sides. 33. Solve a linear equation in one variable that is an identity or a contradiction. 3 1. Simplify an expression using the order of operations and combining like terms 4 Simplifying Expressions The number part of each term is called its coefficient. • The coefficient of 7x is 7. • The coefficient of –3xy is –3. • The coefficient of y2 is the understood factor of 1. • The coefficient of 8 is 8. 5 Simplifying Expressions Like Terms Like terms, or similar terms, are terms with the same variables having the same exponents. Like terms: •3x and 5x •9x2 and –3x2 Unlike terms: •4x and 5x2 6 Simplify an expression using the order of operations and combining like terms Using the distributive property of algebra, combine the terms in 3x + 5x and 9xy2 – 11xy2 . 3x + 5x = (3 + 5)x expressions with like terms = 8x 9xy2 – 11xy2 = (9 – 11)xy2 expressions with = –2xy2 like terms 7 Simplifying Expressions Combining Like Terms To combine like terms, add their coefficients and keep the same variables and exponents. 8 Your Turn Simplify: 3(x + 2) + 2(x – 8) Solution: To simplify the expression, we will use the distributive property to remove parentheses and then combine like terms. 3(x + 2) + 2(x – 8) = 3x + 3 2 + 2x + 2 (–8) = 3x + 6 + 2x – 16 Use the distributive property to remove parentheses. 3 2 = 6 and 2 8 = 16. 9 Your Turn = 3x + 2x + 6 – 16 = 5x – 10 cont’d Use the commutative property of addition: 6 + 2x = 2x + 6. Combine like terms. 10 Simplifying Expressions Comment You a) simplify expression and then b) solve and equation 11 2. Solve a linear equation in one variable requiring simplifying one or both sides 12 Solving an Equation To solve a linear equation in one variable, we must isolate the variable on one side. Solving Equations 1. Clear the equation of any fractions or decimals. 2. Use the distributive property to remove any grouping symbols. 3. Combine like terms on each side of the equation. 13 Solving an Equation 4. Undo the operations of addition and subtraction to collect the variables on one side and the constants on the other. 5. Combine like terms and undo the operations of multiplication and division to isolate the variable. 6. Check the solution in the original equation. 14 Your Turn Solve: 3(x + 2) – 5x = 0. Solution: To solve the equation, we will remove parentheses, combine like terms, and solve for x. 3(x + 2) – 5x = 0 3x + 3 2 – 5x = 0 3x – 5x + 6 = 0 Use the distributive property to remove parentheses. Use the commutative property of addition and simplify. 15 Your Turn -- Solution –2x + 6 = 0 –2x + 6 – 6 = 0 – 6 –2x = –6 cont’d Combine like terms. Subtract 6 from both sides. Combine like terms. Divide both sides by –2. x=3 Simplify. 16 Your Turn – Solution cont’d Check: 3(x + 2) – 5x = 0 3(3 + 2) – 5 3 ≟ 0 35–53≟0 15 – 15 ≟ 0 0=0 Substitute 3 for x. Perform the operation inside the parentheses. Multiply. True. Since the solution 3 checks, the solution set is {3}. 17 3. Solve a linear equation in one variable that is an identity or a contradiction 18 Solving an Equation Conditional Equation: Given: 3x + 1 = 0 is true only when: x = -1/3 Identity: Given: 2x + x = 3x Contradiction: Given: x = x + 1 is true for all R is never true--no solution Solution set = Ø (Empty set) 19 Solving an Equation Three possible types of equations. 20 Your Turn Solve: 3(x + 8) + 5x = 2(12 + 4x). Solution: To solve this equation, we will remove parentheses, combine terms, and solve for x. 3(x + 8) + 5x = 2(12 + 4x) 3x + 24 + 5x = 24 + 8x 8x + 24 = 24 + 8x Use the distributive property to remove parentheses. Combine like terms. 21 Your Turn – Solution 8x + 24 – 8x = 24 + 8x – 8x 24 = 24 cont’d Subtract 8x from both sides. Combine like terms. Since the result 24 = 24 is true for every number x, every number is a solution of the original equation. This equation is an identity. The solution set is the set of real numbers, 22