2.3 Solving for a Variable
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1 l Locker w LESSON. Solving for a Variable Common Core Mat Standards Te student is expected to: A-CED.4 Rearrange formulas to igligt a quantity of interest, using te same reasoning as in solving an equation. Also A-REI. Matematical Practices MP.8 Patterns Language Objective Explain to a partner ow to solve an equation for a specified variable. Name Class Date. Solving for a Variable Essential Question: How do you rewrite formulas and literal equations? Explore Rearranging Matematical Formulas Resource Locker Literal Equations are equations tat contain two or more variables. Tere are many literal equations in te form of mat, science, and engineering formulas. Tese formulas may seem like tey can only be solved for te variable tat is isolated on one side of te formula. By using inverse operations and te properties of equality, a formula can be rearranged so any variable in te formula can be isolated. It is no different tan ow equations are solved by using inverse operations and te properties of equality. How can you solve te equation 4 = 6x? 4 _ = _ 6x 6 6 Wat is te reason for dividing? Multiplication must be undone to solve for x. Te inverse of multiplication is division. ENGAGE Essential Question: How do you rewrite formulas and literal equations? Use te properties of equality to undo operations to isolate a variable. PREVIEW: LESSON PERFORMANCE TASK View te Engage section online. Discuss various familiar benc mark temperatures, suc as te freezing point of water at F or 0 C and te boiling point of water at 1 F or 100 C. Ten preview te Lesson Performance Task. Hougton Mifflin Harcourt Publising Company Wy divide by 6 To undo multiplying by 6, divide by 6.? = x By rearranging te equation x was isolated and te solution was found. Te matematical formula for te volume of a rectangular prism, B = V or V =, is a literal equation. V represents volume, l represents lengt, w represents widt, and represents eigt. Using inverse operations, te formula can be rearranged to solve for any one of te variables tat migt be unknown. Like solving for x, a formula can be rearranged to isolate a variable. 7 In te formula V =, te variable needs to be isolated. Te operation of multiplication is used in te formula. Te inverse operation, division, sould be used to isolate. Reflect V = = 1. Using te formula for a rectangular prism, rewrite te formula to solve for l. V = _ w = w = l, or w l = w Te formula rearranged in tis way can easily produce te eigt of te rectangular prism, wen te volume, lengt, and widt are known. l w Module 67 Lesson Name Class Date. Solving for a Variable Essential Question: How do you rewrite formulas and literal equations? A-CED.4 Rearrange formulas to igligt a quantity of interest, using te same reasoning as in solving an equation. Also A-REI. Explore Rearranging Matematical Formulas Hougton Mifflin Harcourt Publising Company Resource Literal Equations are equations tat contain two or more variables. Tere are many literal equations in te form of mat, science, and engineering formulas. Tese formulas may seem like tey can only be solved for te variable tat is isolated on one side of te formula. By using inverse operations and te properties of equality, a formula can be rearranged so any variable in te formula can be isolated. It is no different tan ow equations are solved by using inverse operations and te properties of equality. How can you solve te equation 4 = 6x? 4_ = 6x _ Wat is te reason for dividing? Wy divide by? = x By rearranging te equation x was isolated and te solution was found. Te matematical formula for te volume of a rectangular prism, B = V or V =, is a literal equation. V represents volume, l represents lengt, w represents widt, and represents eigt. Using inverse operations, te formula can be rearranged to solve for any one of te variables tat migt be unknown. Like solving for x, a formula can be rearranged to isolate a variable. Multiplication must be undone to solve for x. 6 6 Te inverse of multiplication is division. 6 To undo multiplying by 6, divide by 6. 7 In te formula V =, te variable needs to be isolated. Te operation of is used in te formula. Te inverse operation,, sould be used to isolate. Reflect multiplication division V = = 1. Using te formula for a rectangular prism, rewrite te formula to solve for l. V = w = w _ = l, or w l = w Te formula rearranged in tis way can easily produce te eigt of te rectangular prism, wen te volume, lengt, and widt are known. Module 67 Lesson HARDCOVER PAGES 60 Turn to tese pages to find tis lesson in te ardcover student edition. 67 Lesson.
2 Explain 1 Rearranging Scientific Formulas Use inverse operations to isolate te unknown variable in a scientific formula. Te formula for density is D = _ V. Lead as a very ig density of 11,40 kg/ m. Plastic foam as a very low density of 7 kg/ m. Te formula for density can be rearranged to solve for V, volume or m, mass. Example 1 EXPLORE Rearranging Matematical Formulas A sinker on a fising line is made of lead and as a volume of m. Wat is te mass of te sinker? Te density formula can be rearranged to isolate m, te mass. Te values for volume and density can ten be substituted into te formula to find te mass. D = V DV = ( V ) V DV = m (11,40 kg/m ) ( m ) = m 0.17 kg m Te design for a life preserver requires 0. kilogram of plastic foam to provide proper buoyancy. Wat is te volume of te plastic foam required? Rearrange te density formula to isolate V. D = V (D) V = V V DV = _ DV = D m D _ D V = Now substitute te given values. 0. kg V = _ 7 kg/m V = m Your Turn. For altitudes up to 6,000 feet, te relationsip between ground temperature and atmosperic temperature can be described by te formula t = -0.00a + g, in wic t is te atmosperic temperature in degrees Fareneit, a is te altitude, in feet, at wic te atmosperic temperature is measured, and g is te ground temperature in degrees Fareneit. Determine te altitude in feet wen t is -7. F and g is 60 F. t = -0.00a + g t - g a = = =, Te altitude will be,000 feet. Module 68 Lesson PROFESSIONAL DEVELOPMENT Mat Background Some literal equations are general forms of equations tat students frequently encounter. For instance, te equation x = 1 as te general form ax = b. Te solution of suc an equation represents te solutions of all members of tat class of equations. For instance, x = b_ a gives te solution of all equations of te form ax = b. Restrictions may apply to te letters in a literal equation. One common restriction is tat te coefficient of a variable cannot be 0; oterwise, te variable term drops out of te equation. Hougton Mifflin Harcourt Publising Company Image Credits: Beck Potograpy/Aurora Potos/Corbis INTEGRATE TECHNOLOGY Have students use a calculator to evaluate a common formula to find a value, suc as te volume of a tree-dimensional figure wit given dimensions, or te Celsius temperature equivalent to a given Fareneit temperature. How is te process of rearranging a literal equation similar to solving an equation in one variable? In bot cases, you use inverse operations to isolate a variable. EXPLAIN 1 Rearranging Scientific Formulas INTEGRATE MATHEMATICAL PRACTICES Focus on Mat Connections MP.1 Ask students to list some scientific formulas tey ave used. AVOID COMMON ERRORS Encourage students to read te questions carefully. Some students may not pay attention to wic variable tey are solving for. How can you tell weter a rewritten formula is correct? Coose values for te variables tat make te original formula a true statement. Ten substitute tose values in te rewritten formula to see if it is a true statement. Solving for a Variable 68
3 EXPLAIN Rearranging Literal Equations Explain Rearranging Literal Equations Using inverse operations to rearrange literal equations can be applied to any formula. Te interest formula, I = prt, is anoter example of a literal equation. In te formula, I represents interest, p te principal or te initial amount to wic interest will be applied, r te rate at wic interest will be paid, and t is te time in years. Example CONNECT VOCABULARY Te term literal means expressed by letters. A literal equation contains two or more variables tat are represented by letters. Contrast tis meaning to te more common use of te word literal to mean according to te strict meaning of a word; not figurative. How does te solution of a literal equation differ from te solution of a specific equation of te same form? Unlike a specific equation, te literal equation s solution is not a number; it is an expression tat involves te letters representing te coefficients and constants in te literal equation. ELABORATE INTEGRATE MATHEMATICAL PRACTICES Focus on Patterns MP.8 Point out tat a literal equation can be used to provide a rule for calculating te solution of any specific equation aving te same form. For example, wen te literal equation ax + b = c (were a 0) is c - b solved for x, te result is x = _ a. You can substitute values for a, b, and c into tis rule to solve any equation of te form ax + b = c witout going troug te steps of solving te specific equation. SUMMARIZE THE LESSON How do you solve a formula for a variable? You isolate te variable by using properties of equality. Make sure to do te same ting to bot sides of te equation to maintain balance. Hougton Mifflin Harcourt Publising Company Find te number of years used in te calculation of a $1000 loan at an interest rate of % wit interest totaling $600. Solve te formula for t. I = prt I_ pr = _ prt pr I_ pr = t Substitute te given values. Since te interest rate is %, r = 0.0. $600 $ = t 1 = t So te lengt of time for te loan is 1 years. Determine te interest rate for a $000 loan tat will be paid off in 4 years wit interest totaling $640. In order to find te interest rate, solve te formula for r. I = prt I_ prt = _ pt pt I _ = r pt Now substitute te values and simplify. $640 = r $000 4 ( ) ( ) So te interest rate is Your Turn 0.08 = r 8 % per year.. Te formula y = mx + b is te slope-intercept form of te equation of a line. Solve te equation for m. y = mx + b y - b = mx _ y - b x = m Module 69 Lesson COLLABORATIVE LEARNING Peer-to-Peer Activity Have students work in pairs. Have eac pair work wit te equation v = v 0 + at, or ave tem select a literal equation from a list tat you provide. Bot students sould solve te equation for eac of te variables it contains ( v 0, a, and t). Ten ave te students compare teir solutions to make sure tey are te same. If not, tey sould ceck eac oter s work to see were a mistake was made. 69 Lesson.
4 Elaborate 4. Discussion Wat could be a reason for isolating a variable in a literal equation? Possible answer: A reason could be to rewrite an equation to make it easier to evaluate an unknown variable. EVALUATE. Describe a situation in wic a formula could be used more easily if it were rearranged. Include te formula in your description. Possible answer: Te formula d = rt is more useful in te form d r = t wen you need to determine te time it takes to travel a certain distance at a certain speed. 6. Essential Question Ceck-In How do you isolate a variable? Possible answer: Isolate a variable by using te properties of equality and inverse operations. Evaluate: Homework and Practice Online Homework Solve for te indicated variable in eac matematical formula. Hints and Help Extra Practice 1. C = πr for r. A = 1 b for b. y = mx + b for x C π = πr _ π (A) = _ ( 1_ b y = mx + b ) C π = r A = b y - b = mx y - b m = x 4. A = 1 (a + b) for. V = πr for 6. SA = πr + πr for 1_ A = (a + b) A = (a + b) A (a + b) = V = πr V πr = Solve for te indicated variable in eac scientific formula. 7. d = rt for t 8. PV = nrt for T d_ r = t PV nr = T FV - OV 9. A = for OV 10. C = _ Wtc T 1000 for W FV - OV A = T AT = FV - OV AT - FV = -OV FV - AT = OV Wtc C = C = Wtc 1000C tc = W SA = πr + πr SA - πr = πr SA - πr = πr Module 70 Lesson Exercise Dept of Knowledge (D.O.K.) Matematical Practices Hougton Mifflin Harcourt Publising Company ASSIGNMENT GUIDE Concepts and Skills Explore Rearranging Matematical Formulas Example 1 Rearranging Scientific Formulas Example Rearranging Literal Equations CURRICULUM INTEGRATION Practice Exercises 1 6,, Exercises 7 10, Exercises 11 1, 4 Have students complete te table below by writing a formula tat is used in eac subject. Ten ask te students to solve eac formula for eac of its variables. Subject Geometry Personal finance Pysical science Common Formulas Formula AVOID COMMON ERRORS Make sure students ceck tat tey solved for te correct variable. After finding a solution, ave students go back to te original question to make sure tey answered te question fully Recall of Information MP. Reasoning 1 Strategic Tinking MP.4 Modeling Strategic Tinking MP. Reasoning 4 Strategic Tinking MP.4 Modeling Strategic Tinking MP. Reasoning Solving for a Variable 70
5 How are te properties of equality used wen solving for a specified variable? Te properties are used to move numbers from one side of te equation to anoter by adding, subtracting, multiplying, or dividing te same number on bot sides of te equation. Of te four properties of equality, wic two properties of equality ave restrictions? Explain. Te Multiplication and Division Properties of Equality ave restrictions; te numbers tat you multiply or divide by may not equal zero. INTEGRATE TECHNOLOGY Have students coose values for te variables tey are not solving for in a literal equation. Using tese values, ave students use a calculator to find te value of te variable tey are solving for. Ten ave tem substitute all te values in te original literal equation to ceck teir answers. JOURNAL In teir journals, ave students describe a problem tat can be solved more easily if tey first solve a literal equation for a specified variable. Ten ave tem summarize te steps for solving te literal equation. Hougton Mifflin Harcourt Publising Company Image Credits: Ocean/ Corbis Solve for te indicated variable in eac literal equation. 11. p + r = q for p = xy + z for x p = q - r p = q - r _ - 4 = k for j 1. j - 4 = k j - 4 = j k (z + 1) = y for z 18. g ( + z + 1 = _ y z = _ y - 1 a b = c for b x - g = a for x 16. p + 9c = p for c ) = 1 for 19. a(n - ) + 8 = bn for n + _ g an a + 8 = bn 0. Wic is a possible way to rewrite te equation y = x + b to solve for b? A. b = _ y - x C. b = y - x z = xy z y = x x_ = a + g B. b = (y - x) D. b = x(y - ) x = (a + g) = 1_ Coice A, b = y - x, is te correct answer. = 1_ g - _ a = cb a_ c = b 9c = p - p 9c = - 4p c = - 4_ 9 p a + 8 = bn an a + 8 = n (b a) _ a + 8 b a = n _ 8 a b a = n 1. Sports To find a baseball pitcer s earned run average (ERA), you can use te formula Ei = 9r, in wic E represents ERA, i represents te number of innings pitced, and r represents te number of earned runs allowed. Solve te equation for E. Wat is a pitcer s ERA if e allows earned runs in 18 innings pitced? Ei = 9r E = _ 9r i E = 9 18 =.. Meteorology For altitudes up to 6,000 feet, te relationsip between ground temperature and atmosperic temperature can be described by te formula t = -0.00a + g, in wic t is te atmosperic temperature in degrees Fareneit, a is te altitude, in feet, at wic te atmosperic temperature is measured, and g is te ground temperature in degrees Fareneit. Solve te equation for a. If te atmosperic temperature is -6. F and te ground temperature is 7 F, wat is te altitude? t - g = -0.00a t - g = a If te atmosperic temperature is -6. F and te ground temperature is 7 F, ten a = =,000 ft Module 71 Lesson 71 Lesson.
6 H.O.T. Focus on Higer Order Tinking. Explain te Error A student was asked to use te formula for te perimeter of a rectangle, P = l + w, to solve for l. Te student came up wit an answer, P - w = l. Wat error did te student make? Explain. Ten solve for l. Te student only rearranged te formula so te variable was on a side by itself but did not simplify furter to remove te coefficient. Te formula for l is l = P - w. 4. Multi-Step Te formula c = p + 1 relates c, te total cost in dollars of osting a birtday party at a skating rink, to p, te number of people attending. If Allie s parents are willing to spend $00 for a party, ow many people can attend? c = p + 1 c - 1 = p c - 1 = p 00-1 p = = 8 = 17 Te number of people tat can attend te party is 17. CONNECT VOCABULARY Note tat te words for Celsius and Fareneit in Spanis are te cognates Celsio and Fareneit. If you rewrite te fraction 9 as a decimal, te formula for converting Celsius temperatures to Fareneit becomes F = 1.8C +. Solve tis equation to isolate C. Is te result equivalent to te result you got wen you used a fraction? F - C = _ 1 _ 1.8 ; yes, because 1.8 = = 9. Multi-Step Te formula for te area of a triangle is A = 1 b, in wic b represents te lengt of te base and represents te eigt. If a triangle as an area of 19 mm and te eigt is 1 mm, wat is te measure of te base? A = 1_ b (19) = b 1 A = b = b A = b Terefore, te measure of te base is mm. Lesson Performance Task Te following table sows te average low temperatures in Fareneit for te city of Boston for several monts during te year. Te formula F = 9 C + allows you to determine te temperature in Fareneit wen given te temperature in Celsius. Mont Temperature in Fareneit January April 41 July 6 October 47 December 8 Temperature in Celsius -.6 a. Use te information given to determine te average low temperatures in Celsius. b. Would it ever be possible for te temperature in Celsius to ave a greater value tan te temperature in Fareneit? Explain wy or wy not. Yes, for any Fareneit temperature less tan -40, te Celsius temperature will ave a value greater tan te temperature in Fareneit. For example, -49 F = -4 C C = (F - ) 9 Module 7 Lesson Hougton Mifflin Harcourt Publising Company Image Credits: Randy Ducaine/Alamy INTEGRATE MATHEMATICAL PRACTICES Focus on Mat Connections MP.1 Explain tat a kelvin is a unit of temperature tat is used in many scientific calculations. Te unit is called a kelvin, not a degree kelvin. Te temperature 0 K is absolute zero, te coldest temperature possible, so tere are no negative values on te kelvin temperature scale. Te formula K = C + 7 is used to convert degrees Celsius to kelvins. EXTENSION ACTIVITY Ask students weter double a Celsius value and ten add 0 is a good sortcut for estimating a temperature in Fareneit. Ten ave students find a similar metod for estimating a temperature in Celsius given te temperature in Fareneit. Students sould find tat te given sortcut is fairly accurate for temperatures between about 1 C and C, and less accurate for more extreme temperatures. A similar metod for estimating a temperature in Celsius given te temperature in Fareneit is subtract 0 from te Fareneit value and ten alve te result. Scoring Rubric points: Student correctly solves te problem and explains is/er reasoning. 1 point: Student sows good understanding of te problem but does not fully solve or explain is/er reasoning. 0 points: Student does not demonstrate understanding of te problem. Solving for a Variable 7
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