Unit 6: Say It with Symbols//Investigations 1 & //Connections Name Class Date Equivalent Expressions & Combining Expressions Connecting Your Knowledge I can determine when algebraic expressions are equivalent and write algebraic expressions in useful equivalent forms. I can combine symbolic expressions using algebraic operations to form new expressions. Math / 30 Reflection / 10 Total / 40 Points For this packet, your goal is to earn 40. 30 will be from the math problems. 10 will be from the reflection section. Look at the point values of the following questions and answer the questions that you feel most comfortable to answer. For the following questions, tell whether you think the two equations are equivalent. Then explain (using mathematical steps, or the Distributive or Commutative Properties) why or why not? 1. -3x + 6 + x YES or NO? Why or why not? 6 = x. 10 x YES or NO? Why or why not? x 10 3. (3x + 4) + (x 3) YES or NO? Why or why not? x + 1 For the following questions, insert parentheses into the left side of the equation to make the statement true (to equal the right side of the equation). 4. 7 + p p = 11p. 7 + p p = 7 6. 7 + p p = 7 + 4p
Use the Distributive Property to write an equivalent expression for the following questions. 7. ( + x)( + 3x) 8. (x + )(x + 3) Solve each equation below AND check your solution. 9. 7x + 1 = 1x + CHECK: x = 10. -3x + = x 10 CHECK: x = 11. 14 3x = 1.x + CHECK: x = 1. 10 3x =.x CHECK: x =
13. Multiple Choice Which statement is false when a, b, and c are different real numbers? F. (a + b) + c = a + (b + c) G. ab = ba H. (ab)c = a(bc) J. a b = b a The following expression represents the area of a rectangle. Draw a divided rectangle for the expression with the lengths and areas labeled. Then write an equivalent expression in expanded form. 14. (x + 1)(x + 6) Divided Rectangle: Expanded Form: The following expression represents the area of a rectangle. Draw a divided rectangle for the expression with the lengths and areas labeled. Then write an equivalent expression in factored form. 1. 3x + 4x Divided Rectangle: Factored Form: 16. Carson s solution for 11x 1 = 30 + x is shown below. Some steps are missing. 11x - 1 = 30 + x 11x = 4 + x 6x = 4 x = 7
a. Fill in the missing steps between the steps already listed above. b. How can you check that x = 7 is actually the correct solution? 17. The writing club wants to publish a book of students short stories, poems, and essays. A member of the club contacts two local printers to get bids on the cost of printing the books. Bid 1: $100 plus $4 per book Bid : $ plus $7 per book a. Write an equation for each bid. b. Use your equations to find the number of books for which the two bids are EQUAL. 18. Sarah invests D dollars in a money-market account that earns 10% interest per year. She does not plan on taking money out during the year. She writes the expression D + 0.10D to represent the amount of money in the account at the end of one year. a. Explain why this expression is correct. b. Write an equivalent expression in factored form. c. Suppose Sarah invests $1,00. How much money will she have in her account at the end of one year?
10 19. The ski club is planning a trip for winter break. They write the equation C = 00 + 10N to estimate the cost in dollars C of the trip for N students. a. Duncan and Corey both use the equation above to estimate the cost for 0 students. Duncan says the cost is $10,00 and Corey says it is $700. Whose estimate is correct? Show your work to prove. b. How do you think Duncan and Corey found such different estimates if they both used the same equation? c. Suppose 0 students go on the trip. What is the cost per student? d. Write an equation for the cost per student S when N students go on the trip. e. Use your equation from part (d) to find the cost per student when 40 students go on the trip.
10 0. In the following graph, line L1 represents the income for selling N soccer balls. Line L represents the expenses for manufacturing N soccer balls. a. What is the start-up expense (the expense before any soccer balls are produced) for manufacturing the soccer balls? Note: The vertical axis is in thousands of dollars. b. What are the expenses AND income for producing 00 soccer balls? 3,000 soccer balls? c. What is the profit for producing and selling 00 soccer balls? 3,000 soccer balls? d. What is the break-even point? Give both the number of soccer balls and sales for this point. e. What is the equation for income? (Use y = mx + b format) f. What is the equation for expenses? (Use y = mx + b format)
g. What is the equation for profit? (Remember profit = income expenses) h. Suppose the manufacturer produces and sells 1,70 soccer balls. Use your equation from part (g) to find the profit. 10 1. The Phillips Concert Hall staff estimates their concession stand profits P C and admission profits P A with the following equations, where x is the number of people attending (in hundreds): P C = 1x 00 P A = 106x x The concession stand profits include revenue from advertising and the sale of food and souvenirs. The admission profits are based on the difference of ticket sales and cost. a. Write an equation for the total profit for P in terms of the number of people x (in hundreds). b. What is the maximum profit? How many people must attend in order to achieve the maximum profit?
Equivalent Expressions & C ombining Expressions: Ref lection In this Investigation, you found different but equivalent expressions to represent a quantity in a relationship. You also combined expressions or substituted an equivalent expression for a quantity to make new expressions. Answer the following questions in complete sentences. (This reflection is worth 10 of your total of 40.) 1. What does it mean to say that two expressions are equivalent? Use an example to demonstrate.. Explain how you can use the Distributive Property to write equivalent expressions. Use an example to demonstrate. 3. Describe a situation in which it is helpful to add expressions to form a new, single expression. Use an example to demonstrate. 4. Describe a situation in which it is helpful to substitute an equivalent expression for a quantity in an equation. Use an example to demonstrate.. What are the advantages and disadvantages of working with one equation rather than two or more equations in a given situation?