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1-1 Using Variables NCSCOS: 1.02 – Use formulas and algebraic expressions, including iterative and recursive forms, to model and solve problems. Obj. 1: Modeling Relationships with Variables If you earn an hourly wage of $6.50 Hours Worked Pay (dollars) 1 6.50 x 1 2 6.50 x 2 3 6.50 x 3 h 6.50 x h your pay is the number of hours you work multiplied by 6.5. In the table at the bottom left, the variable h stands for the number of hours you worked. A variable is a symbol, usually a letter, that represents one of more numbers. The expression 6.50h is an algebraic expression. An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. Algebraic expressions are sometimes called variable expressions. Ex. 1: Writing an Algebraic Expression Write an algebraic expression for each phrase: Seven more than n + “More than” indicates addition. Add the first number 7 to the second number n. The difference of n and seven − “Difference” indicates subtraction. Begin with the first number n. Then subtract the second number 7 The product of seven and n “Product” indicates multiplication. Multiply the first number 7 by the second number n. The quotient of n and seven “Quotient” indicates division. Divide the first number n by the second number 7. Your turn to write an algebraic expression: The t quotient of 4.2 and c minus 15 Remember: . − To translate an English phrase into an algebraic expression, you may need to define one or more variables first. Ex. 2: Writing an Algebraic Expression Define a variable and write an algebraic expression for each phrase. Two times a number plus 5 Relate: two times a number plus 5 Define: Let n = the number. Write: 2 x n + 5 7 less than three times a number Relate: 7 less than three times Define: Let a = the number. Write: 3 x a a number - 7 Your turn to define a variable and write an algebraic expression for each phrase: 9 less than a number The The sum of twice a number and 31 product of one half of a number and one third of the same number Obj. 2: Modeling Relationships with Equations and Formulas You can use algebraic expressions to write an equation. An equation is a mathematical sentence that uses an equal sign. If the equation is true, then the two expressions on either side of the equal sign represent the same value. An equation that contains one or more variables in an open sentence. In everyday language, the word “is” often suggests an equal sign in the associated equation. Ex. 3: Writing an Equation Track One Media sells all CDs for $12 each. Write an equation for the total cost of given number of CDs. Relate: The total cost is 12 times the number of CDs bought. Define: Let n = the number of CDs bought. Let c = the total cost. Write: Suppose c = 12 x n the manager at Track One Media raises the price of each CD to $15. Write an equation to find the cost of n CDs. Suppose the manager at Track One Media uses the equation c = 10.99n. What could this mean? Ex. 4: Real-World Problem Solving Write an equation for the data in the table below: Cost of Purchase Change from $20 $20.00 $0 $19.00 $1.00 $17.50 $2.50 $11.59 $8.41 Relate: Change equals $20.00 minus purchase. Define: Let c = cost of item purchased. cost of Let a = amount of change. Write: a = $20.00 - c Exercises: Practice and Problem Solving Ex. 1: Practice by Example: Write an algebraic expression for each phrase. 1. 4 more than p 2. y minus 12 3. 12 minus m 4. The product of c and 15 5. The quotient of n and 8 6. The quotient of 17 and k 7. 23 less than x 8. The sum of v and 3 Exercises: Practice and Problem Solving Continued Ex. 2: Define a variable and write an expression for each phrase. 9. 2 more than twice a number. 10. A number minus 11 11. 9 minus a number 12. A number divided by 82 13. The product of 5 and a number 14. The sum of 13 and twice a number 15. The quotient of a number and 6 16. The quotient of 11 and a number Exercises: Practice and Problem Solving Continued Ex. 3: Define variables and write an equation to model each situation. 17. The total cost is the number of cans times $0.70. 18. The perimeter of a square equals 4 times the length of a side. 19. The total length of rope, in feet, used to put up tents is 60 times the number of tents. 20. What is the number of slices of pizza left from an 8-slice pizza after you have eaten some slices? Exercises: Practice and Problem Solving Continued Ex. 4: Define variables and write an equation to model the relationship in each table. 21. 23. Number of Workers Number of Radios Built 1 22. Number of Tapes Cost 13 1 $8.50 2 26 2 $17.00 3 39 3 $25.50 4 52 4 $34.00 Number of Sales Total Earnings Number of Hours Total Pay 5 $2.00 4 $32 10 $4.00 6 $48 15 $6.00 8 $64 20 $8.00 10 $80 24. Apply Your Skills: Write an expression for each phrase. 25. 26. 27. 28. 29. 30. 31. 32. 33. The sum of 9 and k minus 17 6.7 more than 5 times n 9.85 less than the product of t and 37 The quotient of 3b and 4.5 15 plus the quotient of 60 and w 7 minus the product of v and 3 The product of m and 5, minus the quotient of t and 7 The sum of the quotient of p and 14 and the quotient of q and 3 8 minus the product of 9 and r Write a phrase for each expression. 34. q+5 35. 3–t 36. 9n + 1 37. 5 38. 7hb Define variables and write an equation to model the relationship in each table. 39. Number of Days Change in Height (meters) 1 40. Time (months) Length (inches) 0.165 1 4.1 2 0.330 2 8.2 3 0.495 3 12.3 4 0.660 4 16.4 Use the table below: Lawns Mowed Hours 1 2 3 6 Does each statement fit the data in the table? Explain. Hours worked = lawns mowed x 2 Hours worked = lawns mowed + 3 Challenge: The table at the right shows the height of the first bounce when a ball is dropped from different heights. Write an equation to describe the relationship between the height of the first bounce and the drop height. Suppose Drop Height (ft.) Height of First Bounce (ft.) 1 1 2 2 1 3 1 4 5 1 2 2 2 1 2 you drop the ball from a window 20 ft. above the ground. Predict how high the ball will bounce. Open-Ended Describe a real-world situation that each equation could represent. Include a definition for each variable. d = 5t a =b+3 c = 40 ℎ Standardized Test Prep Which is an algebraic expression for “six less than k”? 1. a. b. 2. 6 6 6–k d. k – 6 Which is an algebraic expression for “the product of a and 10”? a. a + 10 b. a – 10 c. 10a c. d. 3. 10 4. Which is an algebraic expression for “9 more than v”? a. v + 9 b. v– 9 c. 9 – v d. 9v A container of milk contains 64 ounces. Which equation models the number n of ounces remaining after you have drunk m ounces? a. − 64 = b. 64 − = c. − 64 = d. − = 64 Standardized Test Prep 5. Which equation models the relationship in the table if r represents the row number and t represents the number of tulips? a. = 3 Row Number Number of Tulips b. =3 1 3 2 6 3 9 =+3 4 12 d. = 3 Which is an algebraic expression for “the quotient of r + 5 and b”? c. 6. a. b. +5 +5 c. d. +5 +5 Add, subtract, multiply, or divide. 12. 0.2 + 0.7 0.13 + 0.91 0.6 + 0.75 1.09 + 0.37 0.9 x 0.07 0.58 – 0.49 19. List four prime numbers between 20 and 50. 7. 8. 9. 10. 11. 13. 14. 15. 16. 17. 18. 0.8 – 0.66 1.32 – 0.39 2 x 0.5 0.69 ÷ 3 0.6 ÷ 0.2 1.21 ÷ 11