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Bails – Math 094 Notes Section 2.1 Introduction to Variables Variable: letter that represents part of a rule that varies. Constant: part of a rule that does not change. Coefficient: the number in front of the variable. Expression: combination of variables and constants terms separated by operations. For the following expressions, identify the variable term, coefficient, and constant term. 1. 3x 7 variable term = coefficient = constant term = 2. 4y 11 variable term = coefficient = constant term = 3. x 6 variable term = coefficient = constant term = 4. y variable term = coefficient = constant term = 5. 30 variable term = coefficient = constant term = Review of Exponents: Recall 32 = 3 3 = 9 so 3x2 = 3x x and Rewrite each expression without exponents. 6. 7. 2x 3 8. x4y 3 9. x2y3 = x x y y y xy 3 9x 2 yz3 Steps for evaluating an expression: 1. Write the original expression 2. Copy the original again, but replace the _______________ with a set of parenthesis 3. Place given value of variable inside parenthesis 4. Simplify Evaluate each expression when a 3 , b 1, and c 2 3abc 10. 11. a3 21 Chapter 2 Understanding Variables and Solving Equations Evaluate each expression when a 3 , b 1, and c 2 12. 2a2 b2 13. ab bc 14. 2a c b 15. 3c 2 5b 3 a 16. The expression for determining the cost per ounce is c z where c is the total cost and z is the number of ounces. Evaluate the expression when the total cost of caviar is $48 for 16 oz. 17. The expression for determining the perimeter of a rectangle is 2L + 2W, where L is the length and W is the width. Evaluate the expression when a. The length is 15 centimeters (cm) and the width is 11 centimeters. b. The length is 20 feet (ft) and the width is 15 feet. 22 Bails – Math 094 Notes Section 2.2 Simplifying Expressions Similar (Like) Terms: have the same variable(s) and exponents. The number in front of the variable (coefficient) can differ. Like Terms Examples: 3x, 4x 2xy, -5xy 4y2, 10y2 Unlike Terms Examples: 3x, 3y 3x2, 4x 4x, 5xy Steps for simplifying an expression: 1. Rewrite expression without parenthesis If necessary, make sure to distribute negative through parentheses 2. Identify like ______________________ 3. Add/subtract the ______________________ (number in front of variable) & copy the variable **Variables always stay the same in addition and subtraction 4. Write each answer with the variables in alphabetical order and any constant term last. Simplify (combine like terms) if possible. 3x 8x 1. 2. xxx 3. 3y 7y y 4. 2x 2 5 x 2 5. 8 xy 9xy xy 6. 2x 3 4 x 1 7. 3x 5 9x 2 8. 4y 5 x 3y 7 x 9. 3 x 2 4 x x 2 11x 10. 6x 3 7x 2 10x 23 Chapter 2 Understanding Variables and Solving Equations Review of Associative Property of Multiplication: (a b) c = a (b c) Simplify by using the associative property of multiplication. 11. 8 3x = 12. 9 3x 2 = Review of Distributive Property: a b c ab ac Note: If there is no number (coefficient) in front of the variable, the number is 1. Therefore, x = 1x Use the distributive property to simplify each expression. 13. 2 6 x 1 = 14. 3 x 2 = 15. 4 4 x 1 = 16. 5 7 x 3 = 18. 3 x 2 15 Simplify each expression. 2 6x 5 2 17. 19. 4 6 9x 2 20. 6 10 x 1 21. 16 x 6 9 x 2 18 22. 3 5 2x 7 15 x 23. 2 3 x 7 4 8 x 1 24. 5 x 3 2x 9 4 3 x 2 11 24 Bails – Math 094 Notes Section 2.3 Solving Equations Using Addition Addition Property of Equality If A=B Then A+C=B+C A, B, C are algebraic expressions Adding/subtracting the same number to both sides of an equation keeps the equation balanced. Note: A solution for an equation is a number that makes the equation a true statement. Solve each equation and check the answer. 1. x 7 11 2. x 7 11 3. x 7 11 4. x 7 11 5. 33 27 x 6. 52 x 61 7. x 17 21 28 8. 16 x 30 14 25 Chapter 2 Understanding Variables and Solving Equations For each equation, simplify each side (if possible) then solve and check the answer. 9. x 8 4 11 9 10. 6x 4 5x 3 11. 7 x 8 6 x 3 1 12. 3 6 9 9x 5 8x 13. 6 x 7 x 9 19 8 14. x 3 7 x 9 x 4 7 9 26 Bails – Math 094 Notes Section 2.4 Solving Equations Using Division Division Property of Equality If A = B, then A B C C as long as C 0 Dividing by the same nonzero number on both sides of an equation keeps the equation balanced. Solve each equation and check the answer. 1. 6 x 42 2. 5 x 40 3. 8 x 48 4. 121 11x 5. 27 x 0 6. 14 x For each equation, simplify each side (if possible) then solve and check the answer. 7. 2 x 3 x 17 23 8. 8 x 15 x 71 15 27 Chapter 2 Understanding Variables and Solving Equations For each equation, simplify each side (if possible) then solve and check the answer. 9. 8 2x 32 10. 60 2 15x 11. 14 x 10 x 36 44 12. 7 x 5 x 7 4 59 13. 82 96 5 4 x 8 2x 3 x 14. 36 36 41 50 x 3 2x 4 4 x 28 Bails – Math 094 Notes Section 2.5 Solving Equations with Several Steps Steps for solving equations with variables on both sides: 1. Simplify each side of equation by removing parentheses and combining like terms 2. Use _____________________ property to move all variables (letters) to one side. 3. Use _____________________ property to move all constant terms (numbers) to the opposite side of the equation. 4. Use division property to get a _____________________ (the number in front of the variable) of 1. 5. Check the solution by going back to the original equation. Solve each equation and show all work. 1. 5 x 10 55 2. 7 x 4 31 3. 3 x 6 36 4. 42 3 x 7 5. 3 x 2 x 40 6. 11x 4 x 8 15 29 Chapter 2 Understanding Variables and Solving Equations Solve each equation and show all work. 7. 5 x 3 2x 3 8. 2x 7 4x 1 9. 5 x 10 19 8 x 10. 15 x 1 4 x 20 11. 8 2x 1 6 3 x 5 12. 7 x 8 2 x 13 13. 12 x 2 5 2x 1 14. x 28 10 4 x 6 27 30