2.4. Algebraic Reasoning. Essential Question How can algebraic properties help you solve an equation?

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1 2.4 TEXS ESSENTIL KNOWLEGE N SKILLS Preparing for G.6. G.6. G.6. G.6.E lgebraic Reasoning Essential Question How can algebraic properties help you solve an equation? Justifying Steps in a Solution Work with a partner. In previous courses, you studied different properties, such as the properties of equality and the istributive, ommutative, and ssociative Properties. Write the property that justifies each of the following solution steps. lgebraic Step Justification 2(x + 3) 5 = 5x + 4 Write given equation. 2x = 5x + 4 2x + = 5x + 4 2x 2x + = 5x 2x + 4 = 3x = 3x = 3x 3 3 = 3x 3 = x x = NLYZING MTHEMTIL RELTIONSHIPS To be proficient in math, you need to look closely to discern a pattern or structure. Stating lgebraic Properties Work with a partner. The symbols and represent addition and multiplication (not necessarily in that order). etermine which symbol represents which operation. Justify your answer. Then state each algebraic property being illustrated. Example of Property Name of Property 5 6 = = (5 6) = (4 5) 6 4 (5 6) = (4 5) = = 5 5 = 5 4 (5 6) = ommunicate Your nswer 3. How can algebraic properties help you solve an equation? 4. Solve 3(x + ) = 3. Justify each step. Section 2.4 lgebraic Reasoning 9

2 2.4 Lesson What You Will Learn ore Vocabulary Previous equation solve an equation formula Use lgebraic Properties to justify the steps in solving an equation. Use the istributive Property to justify the steps in solving an equation. Use properties of equality involving segment lengths and angle measures. Using lgebraic Properties When you solve an equation, you use properties of real numbers. Segment lengths and angle measures are real numbers, so you can also use these properties to write logical arguments about geometric figures. ore oncept lgebraic Properties Let a, b, and c be real numbers. ddition Property If a = b, then a + c = b + c. Subtraction Property If a = b, then a c = b c. Multiplication Property If a = b, then a c = b c, c 0. ivision Property If a = b, then a c = b c, c 0. Substitution Property If a = b, then a can be substituted for b (or b for a) in any equation or expression. REMEMER Inverse operations undo each other. ddition and subtraction are inverse operations. Multiplication and division are inverse operations. Justifying Steps Solve 3x + 2 = 23 4x. Justify each step. 3x + 2 = 23 4x Write the equation. Given 3x x = 23 4x + 4x dd 4x to each side. ddition Property 7x + 2 = 23 ombine like terms. Simplify. 7x = 23 2 Subtract 2 from each side. Subtraction Property 7x = 2 ombine constant terms. Simplify. x = 3 ivide each side by 7. ivision Property The solution is x = 3. Monitoring Progress Help in English and Spanish at igideasmath.com Solve the equation. Justify each step.. 6x = p 9 = 0p z = + 5z 92 hapter 2 Reasoning and Proofs

3 Using the istributive Property ore oncept istributive Property Let a, b, and c be real numbers. Sum a(b + c) = ab + ac ifference a(b c) = ab ac Using the istributive Property Solve 5(7w + 8) = 30. Justify each step. 5(7w + 8) = 30 Write the equation. Given 35w 40 = 30 Multiply. istributive Property 35w = 70 dd 40 to each side. ddition Property w = 2 ivide each side by 35. ivision Property The solution is w = 2. Solving a Real-Life Problem You get a raise at your part-time job. To write your raise as a percent, use the formula p(r + ) = n, where p is your previous wage, r is the percent increase (as a decimal), and n is your new wage. Solve the formula for r. What is your raise written as a percent when your hourly wage increases from $7.25 to $7.54 per hour? REMEMER When evaluating expressions, use the order of operations. Step Solve for r in the formula p(r + ) = n. p(r + ) = n Write the equation. Given pr + p = n Multiply. istributive Property pr = n p Subtract p from each side. Subtraction Property r = n p ivide each side by p. ivision Property p Step 2 Evaluate r = n p when n = 7.54 and p = p r = n p p Your raise is 4% = 7.25 = = 0.04 Monitoring Progress Solve the equation. Justify each step. Help in English and Spanish at igideasmath.com 4. 3(3x + 4) = = 0b + 6(2 b) 6. Solve the formula = bh for b. Justify each step. Then find the base of a 2 triangle whose area is 952 square feet and whose height is 56 feet. Section 2.4 lgebraic Reasoning 93

4 Using Other Properties The following properties of equality are true for all real numbers. Segment lengths and angle measures are real numbers, so these properties of equality are true for all segment lengths and angle measures. ore oncept Reflexive, Symmetric, and Transitive Properties Real Numbers Segment Lengths ngle Measures Reflexive Property a = a = m = m Symmetric Property If a = b, then b = a. If =, then =. If m = m, then m = m. Transitive Property If a = b and b = c, then a = c. If = and = EF, then = EF. If m = m and m = m, then m = m. You reflect the beam of a spotlight off a mirror lying flat on a stage, as shown. etermine whether m = m E. Using Properties with ngle Measures E 2 3 m = m 3 Marked in diagram. Given m = m 3 + m 2 dd measures of ngle ddition Postulate (Post..4) adjacent angles. m = m + m 2 Substitute m Substitution Property for m 3. m + m 2 = m E dd measures of ngle ddition Postulate (Post..4) adjacent angles. m = m E oth measures are Transitive Property equal to the sum m + m 2. Monitoring Progress Help in English and Spanish at igideasmath.com Name the property of equality that the statement illustrates. 7. If m 6 = m 7, then m 7 = m = m = m 2 and m 2 = m 5. So, m = m hapter 2 Reasoning and Proofs

5 Modeling with Mathematics park, a shoe store, a pizza shop, and a movie theater are located in order on a city street. The distance between the park and the shoe store is the same as the distance between the pizza shop and the movie theater. Show that the distance between the park and the pizza shop is the same as the distance between the shoe store and the movie theater.. Understand the Problem You know that the locations lie in order and that the distance between two of the locations (park and shoe store) is the same as the distance between the other two locations (pizza shop and movie theater). You need to show that two of the other distances are the same. 2. Make a Plan raw and label a diagram to represent the situation. park shoe store pizza shop movie theater Modify your diagram by letting the points P, S, Z, and M represent the park, the shoe store, the pizza shop, and the movie theater, respectively. Show any mathematical relationships. P S Z M Use the Segment ddition Postulate (Postulate.2) to show that PZ = SM. 3. Solve the Problem PS = ZM Marked in diagram. Given PZ = PS + SZ dd lengths of Segment ddition Postulate (Post..2) adjacent segments. SM = SZ + ZM dd lengths of Segment ddition Postulate (Post..2) adjacent segments. PS + SZ = ZM + SZ dd SZ to each side ddition Property of PS = ZM. PZ = SM Substitute PZ for Substitution Property PS + SZ and SM for ZM + SZ. 4. Look ack Reread the problem. Make sure your diagram is drawn precisely using the given information. heck the steps in your solution. Monitoring Progress Help in English and Spanish at igideasmath.com Name the property of equality that the statement illustrates. 0. If JK = KL and KL = 6, then JK = 6.. PQ = ST, so ST = PQ. 2. ZY = ZY 3. In Example 5, a hot dog stand is located halfway between the shoe store and the pizza shop, at point H. Show that PH = HM. Section 2.4 lgebraic Reasoning 95

6 2.4 Exercises ynamic Solutions available at igideasmath.com Vocabulary and ore oncept heck. VOULRY The statement The measure of an angle is equal to itself is true because of what property? 2. IFFERENT WORS, SME QUESTION Which is different? Find both answers. What property justifies the following statement? If c = d, then d = c. If JK = LM, then LM = JK. If e = f and f = g, then e = g. If m R = m S, then m S = m R. Monitoring Progress and Modeling with Mathematics In Exercises 3 and 4, write the property that justifies each step. 3. 3x 2 = 7x + 8 Given 4x 2 = 8 4x = 20 x = (x ) = 4x + 3 Given 5x 5 = 4x + 3 x 5 = 3 x = 8 In Exercises 5 4, solve the equation. Justify each step. (See Examples and 2.) 5. 5x 0 = x + 7 = x 8 = 6x x + 9 = 6 3x 9. 5(3x 20) = (2x + ) = 9. 2( x 5) = (3x + 4) = 8x 3. 4(5x 9) = 2(x + 7) 4. 3(4x + 7) = 5(3x + 3) In Exercises 5 20, solve the equation for y. Justify each step. (See Example 3.) 5. 5x + y = x + 2y = y + 0.5x = x 3 4 y = y = 30x x + 7 = 7 + 9y In Exercises 2 24, solve the equation for the given variable. Justify each step. (See Example 3.) 2. = 2πr ; r 22. I = Prt; P 23. S = 80(n 2); n 24. S = 2πr 2 + 2πrh; h In Exercises 25 32, name the property of equality that the statement illustrates. 25. If x = y, then 3x = 3y. 26. If M = M, then M + 5 = M x = x 28. If x = y, then y = x. 29. m Z = m Z 30. If m = 29 and m = 29, then m = m. 3. If = LM, then LM =. 32. If = XY and XY = 8, then = hapter 2 Reasoning and Proofs

7 In Exercises 33 40, use the property to copy and complete the statement. 33. Substitution Property : If = 20, then + =. 34. Symmetric Property : If m = m 2, then. 35. ddition Property : If =, then + EF =. 36. Multiplication Property : If =, then 5 =. 37. Subtraction Property : If LM = XY, then LM GH =. 38. istributive Property: If 5(x + 8) = 2, then + = Transitive Property : If m = m 2 and m 2 = m 3, then. 40. Reflexive Property : m =. ERROR NLYSIS In Exercises 4 and 42, describe and correct the error in solving the equation x = x x = 24 x = 3 6x + 4 = 32 6x = 8 x = 3 Given ddition Property ivision Property Given ivision Property Simplify. 43. REWRITING FORMUL The formula for the perimeter P of a rectangle is P = 2 + 2w, where is the length and w is the width. Solve the formula for. Justify each step. Then find the length of a rectangular lawn with a perimeter of 32 meters and a width of 5 meters. 44. REWRITING FORMUL The formula for the area of a trapezoid is = 2 h (b + b 2 ), where h is the height and b and b 2 are the lengths of the two bases. Solve the formula for b. Justify each step. Then find the length of one of the bases of the trapezoid when the area of the trapezoid is 9 square meters, the height is 7 meters, and the length of the other base is 20 meters. 45. NLYZING RELTIONSHIPS In the diagram, m = m E. Show that m = m 3. (See Example 4.) NLYZING RELTIONSHIPS In the diagram, =. Show that =. (See Example 5.) E 47. NLYZING RELTIONSHIPS opy and complete the table to show that m 2 = m 3. E Equation m = m 4, m EHF = 90, m GHF = 90 m EHF = m GHF m EHF = m + m 2 m GHF = m 3 + m 4 m + m 2 = m 3 + m 4 m 2 = m H F 4 Given G Reason Substitution Property 48. WRITING ompare the Reflexive Property with the Symmetric Property. How are the properties similar? How are they different? Section 2.4 lgebraic Reasoning 97

8 RESONING In Exercises 49 and 50, show that the perimeter of is equal to the perimeter of MTHEMTIL ONNETIONS In the figure, ZY XW, ZX = 5x + 7, YW = 0 2x, and YX = 3. Find ZY and XW. Z V Y X 52. HOW O YOU SEE IT? The bar graph shows the number of hours each employee works at a grocery store. Give an example of the Reflexive, Symmetric, and Transitive Properties. Hours Hours Worked Employee 6 7 W 53. TTENING TO PREISION Which of the following statements illustrate the Symmetric Property of Equality? Select all that apply. If = RS, then RS =. If x = 9, then 9 = x. E F If =, then =. = If = LM and LM = RT, then = RT. If XY = EF, then FE = XY. 54. THOUGHT PROVOKING Write examples from your everyday life to help you remember the Reflexive, Symmetric, and Transitive Properties. Justify your answers. 55. MULTIPLE REPRESENTTIONS The formula to convert a temperature in degrees Fahrenheit ( F) to degrees elsius ( ) is = 5 (F 32). 9 a. Solve the formula for F. Justify each step. b. Make a table that shows the conversion to Fahrenheit for each temperature: 0, 20, 32, and 4. c. Use your table to graph the temperature in degrees Fahrenheit as a function of the temperature in degrees elsius. Is this a linear function? 56. RESONING Select all the properties that would also apply to inequalities. Explain your reasoning. ddition Property Subtraction Property Substitution Property Reflexive Property E Symmetric Property F Transitive Property Maintaining Mathematical Proficiency Name the definition, property, or postulate that is represented by each diagram. (Section.2, Section.3, and Section.5) 57. X 58. M Y Reviewing what you learned in previous grades and lessons Z L P N m + m = m XY + YZ = XZ 98 hapter 2 Reasoning and Proofs