1.4 Rewriting Equations And Formulas - Coppin Academy
Page 1 of 21.4Rewriting Equationsand FormulasWhat you should learnGOAL 1 Rewrite equationswith more than one variable.GOAL 2 Rewrite commonformulas, as applied inExample 5.Why you should learn itREFE To solve real-lifeproblems, such as findinghow much you should chargefor tickets to a benefit concert in Example 4. AL LIGOAL 1EQUATIONS WITH MORE THAN ONE VARIABLEIn Lesson 1.3 you solved equations with one variable. Many equations involve morethan one variable. You can solve such an equation for one of its variables.Rewriting an Equation with More Than One VariableEXAMPLE 1Solve 7x º 3y 8 for y.SOLUTION7x º 3y 8Write original equation.º3y º7x 87383y x º Subtract 7x from each side.Divide each side by º3.ACTIVITYDevelopingConceptsEquations with More Than One VariableGiven the equation 2x 5y 4, use each method below to find y whenx º3, º1, 2, and 6. Tell which method is more efficient.Method 1Substitute x º3 into 2x 5y 4 and solve for y. Repeat thisprocess for the other values of x.Method 2Solve 2x 5y 4 for y. Then evaluate the resulting expressionfor y using each of the given values of x.EXAMPLE 2Calculating the Value of a VariableGiven the equation x xy 1, find the value of y when x º1 and x 3.SOLUTIONSolve the equation for y.x xy 1Write original equation.xy 1 º xSubtract x from each side.1ºxxy Divide each side by x.Then calculate the value of y for each value of x.1 º (º1)º1When x º1: y º226Chapter 1 Equations and Inequalities1º3323When x 3: y º
Page 1 of 2REFELAL IBenefit ConcertEXAMPLE 3Writing an Equation with More Than One VariableYou are organizing a benefit concert. You plan on having only two types of tickets:adult and child. Write an equation with more than one variable that represents therevenue from the concert. How many variables are in your equation?SOLUTIONPROBLEMSOLVINGSTRATEGYTotal AdultChild Number Number ofrevenueticket priceof adultsticket pricechildrenVERBALMODELLABELSALGEBRAICMODELTotal revenue R(dollars)Adult ticket price p1(dollars per adult)Number of adults A(adults)Child ticket price p2(dollars per child)Number of children C(children)R p1 A p2 CThis equation has five variables. The variables p1 and p2 are read as “p sub one” and“p sub two.” The small lowered numbers 1 and 2 are subscripts used to indicate thetwo different price variables.EXAMPLE 4Using an Equation with More Than One VariableBENEFIT CONCERT For the concert in Example 3, your goal is to sell 25,000 intickets. You plan to charge 25.25 per adult and expect to sell 800 adult tickets. Youneed to determine what to charge for child tickets. How much should you charge perchild if you expect to sell 200 child tickets? 300 child tickets? 400 child tickets?FOCUS ONAPPLICATIONSSOLUTIONFirst solve the equation R p1A p2C from Example 3 for p2.R p1A p2CWrite original equation.R º p1 A p 2 CSubtract p1A from each side.R º p1A p2CDivide each side by C.Now substitute the known values of the variables into the equation.REFELAL IBENEFITCONCERTFarm Aid, a type of benefitconcert, began in 1985.Since that time Farm Aidhas distributed more than 13,000,000 to family farmsthroughout the UnitedStates.25,000 º 25.25(800)200If C 200, the child ticket price is p2 24.25,000 º 25.25(800)300If C 300, the child ticket price is p2 16.25,000 º 25.25(800)400If C 400, the child ticket price is p2 12.1.4 Rewriting Equations and Formulas27
Page 1 of 2GOAL 2REWRITING COMMON FORMULASThroughout this course you will be using many formulas. Several are listed below.COMMON FORMULASFORMULAVARIABLESDistanced rtd distance, r rate, t timeSimple InterestI PrtI interest, P principal, r rate, t timeTemperatureF C 32Area of Triangle95F degrees Fahrenheit, C degrees CelsiusA bh12A area, b base, h heightArea of RectangleA wA area, length, w widthPerimeter of RectangleP 2 2wP perimeter, length, w widthArea of TrapezoidA (b1 b2)hA area, b1 one base, b2 other base, h heightArea of CircleA πr 2A area, r radiusCircumference of CircleC 2πrC circumference, r radius12EXAMPLE 5The formula for the perimeter of a rectangle is P 2 2w. Solve for w.STUDENT HELPSkills ReviewFor help with perimeter,see p. 914.FELAL IRERewriting a Common FormulaGardeningSOLUTIONP 2 2wWrite perimeter formula.P º 2 2wSubtract 2 from each side.P º 2 w2Divide each side by 2.EXAMPLE 6Applying a Common FormulaYou have 40 feet of fencing with which to enclose a rectangular garden. Express thegarden’s area in terms of its length only.SOLUTIONUse the formula for the area of a rectangle, A w, and the result of Example 5.A wWrite area formula.A P º 2 Substitute }} for w.2 P º2 2 40 º 2 A 2A (20 º )28Chapter 1 Equations and InequalitiesSubstitute 40 for P.Simplify.
Page 1 of 2GUIDED PRACTICE Concept Check Vocabulary Check?.1. Complete this statement: A w is an example of a(n) 2. Which of the following are equations with more than one variable?A. 2x 5 9 º 5xB. 4x 10y 62C. x º 8 3y 73. Use the equation from Example 3. Describe how you would solve for A.Skill Check Solve the equation for y.4. 4x 8y 175. 5x º 3y 96. 5y º 3x 1537. x 5y 2048. xy 2x 8219. x º y 1232In Exercises 10 and 11, use the following information.The area A of an ellipse is given by the formula A πab where a and b are halfthe lengths of the major and minor axes. (The longer chord is the major axis.)10. Solve the formula for a.b11. Use the result from Exercise 10 to find thelength of the major axis of an ellipse whosearea is 157 square inches and whose minoraxis is 10 inches long. (Use 3.14 for π.)aabPRACTICE AND APPLICATIONSSTUDENT HELPExtra Practiceto help you masterskills is on p. 940.EXPLORING METHODS Find the value of y for the given value of x using twomethods. First, substitute the value of x into the equation and then solve for y.Second, solve for y and then substitute the value of x into the equation.12. 4x 9y 30; x 313. 5x º 7y 12; x 114. xy 3x 25; x 515. 9y º 4x º16; x 816. ºy º 2x º11; x º417. ºx 3y º 55; x 2018. x 24 xy; x º1219. ºxy 3x 30; x 1520. º4x 7y 7 0; x 721. 6x º 5y º 44 0; x 41422. x º y 19; x 6253923. x º y 12; x 10411REWRITING FORMULAS Solve the formula for the indicated variable.24. Circumference of a CircleSTUDENT HELPHOMEWORK HELPExamples 1, 2: Exs. 12–23Examples 3, 4: Exs. 33–39Examples 5, 6: Exs. 24–32,40–42Solve for r: C 2πr25. Volume of a Cone1Solve for h: V πr2h326. Area of a Triangle1Solve for b: A bh227. Investment at Simple Interest28. Celsius to Fahrenheit9Solve for C: F C 32529. Area of a Trapezoid1Solve for b2: A (b1 b2)h2Solve for P: I Prt1.4 Rewriting Equations and Formulas29
Page 1 of 2GEOMETRY CONNECTION In Exercises 30º32, solve the formula for the indicatedvariable. Then evaluate the rewritten formula for the given values. (Includeunits of measure in your answer.)30. Area of a circularring: A 2πpwSolve for p. Find pwhen A 22 cm2and w 2 cm.31. Surface area of a32. Perimeter of a track:cylinder:S 2πrh 2πr 2Solve for h. Find hwhen S 105 in.2and r 3 in.P 2πr 2xSolve for r. Find r whenP 440 yd andx 110 yd.rrrhxHONEYBEES In Exercises 33 and 34, use the following information.A forager honeybee spends about three weeks becoming accustomed to the immediatesurroundings of its hive and spends the rest of its life collecting pollen and nectar. Thetotal number of miles T a forager honeybee flies in its lifetime L (in days) can bemodeled by T m(L º 21) where m is the number of miles it flies each day.33. Solve the equation T m(L º 21) for L.34. A forager honeybee’s flight muscles last only about 500 miles; after that the beedies. Some forager honeybees fly about 55 miles per day. Approximately howmany days do these bees live?FOCUS ONCAREERSBASEBALL In Exercises 35 and 36, use the following information.The Pythagorean Theorem of Baseball is a formula for approximating a team’s ratioof wins to games played. Let R be the number of runs the team scores during theseason, A be the number of runs allowed to opponents, W be the number of wins,and T be the total number of games played. Then the formulaWR2 2TR A2approximates the team’s ratio of wins to games played. Source: Inside Sports35. Solve the formula for W.36. The 1998 New York Yankees scored 965 runs and allowed 656. How many of its162 games would you estimate the team won?FUNDRAISER In Exercises 37–39, use the following information.Your tennis team is having a fundraiser. You are going to help raise money by sellingsun visors and baseball caps.REFELAL ISPORTSSTATISTICIANSINTare employed by manyprofessional sports teams,leagues, and news organizations. They collect andanalyze team and individualdata on items such asscoring.NEER TCAREER LINKwww.mcdougallittell.com3037. Write an equation that represents the total amount of money you raise.38. How many variables are in the equation? What does each represent?39. Your team raises a total of 4480. Give three possible combinations of sun visorsand baseball caps that could have been sold if the price of a sun visor is 3.00and the price of a baseball cap is 7.00.The formula for the area of a circle is A πr 2. Theformula for the circumference of a circle is C 2πr. Write a formula for thearea of a circle in terms of its circumference.40. GEOMETRYCONNECTIONChapter 1 Equations and Inequalities
Page 1 of 2INTSTUDENT HELPNEER TThe formulafor the height h of an equilateral triangle is41. GEOMETRYHOMEWORK HELPCONNECTION 3 2h b where b is the length of a side.Visit our Web sitewww.mcdougallittell.comfor help with problemsolving in Exs. 41 and 42.bbhWrite a formula for the area of an equilateraltriangle in terms of the following.ba. the length of a side onlyb. the height onlyThe surfacearea S of a cylinder is given by the formulaS 2πrh 2πr 2. The height h of thecylinder shown at the right is 5 morethan 3 times its radius r.42. GEOMETRYCONNECTIONrha. Write a formula for the surface area ofthe cylinder in terms of its radius.b. Find the surface area of the cylinder forr 3, 4, and 6.TestPreparationSTUDENT HELPQUANTITATIVE COMPARISON In Exercises 43 and 44, choose the statementthat is true about the given quantities.A¡B¡C¡D¡43.The quantity in column A is greater.The quantity in column B is greater.The two quantities are equal.The relationship cannot be determined from the given information.Column AColumn BV whV wh5 cmSkills ReviewFor help with thePythagorean theorem,see p. 917.7 cm3 cm3 cm4 cm44.7 cmV πr 2hV πr 2h4 in.6 in.6 in.1.4 Rewriting Equations and Formulas4 in.31
Page 1 of 2 Challenge45.FUEL EFFICIENCY The moreaerodynamic a vehicle is, the less fuel thevehicle’s engine must use to overcome airresistance. To design vehicles that are asfuel efficient as possible, automotiveengineers use the formulaR 0.00256 ª DC ª FA ª s2where R is the air resistance (in pounds), DC is the drag coefficient, FA is thefrontal area of the vehicle (in square feet), and s is the speed of the vehicle(in miles per hour). The formula assumes that there is no wind.a. Rewrite the formula to find the drag coefficient in terms of the other variables.EXTRA CHALLENGEb. Find the drag coefficient of a car when the air resistance is 50 pounds, thefrontal area is 25 square feet, and the speed of the car is 45 miles per hour.www.mcdougallittell.comMIXED REVIEWWRITING EXPRESSIONS Write an expression to answer the question.(Skills Review, p. 929)46. You buy x birthday cards for 1.85 each. How much do you spend?47. You have 30 and spend x dollars. How much money do you have left?48. You drive 55 miles per hour for x hours. How many miles do you drive?49. You have 250 in your bank account and you deposit x dollars. How muchmoney do you now have in your account?50. You spend 42 on x music cassettes. How much does each cassette cost?51. A certain ball bearing weighs 2 ounces. A box contains x ball bearings. What isthe total weight of the ball bearings?UNIT ANALYSIS Give the answer with the appropriate unit of measure.(Review 1.1) 7 meters52. (60 minutes)1 minute168 hours53. (52 weeks)1 week1354. 4 feet 7 feet441755. 13 liters º 8 liters483 yards56. (12 seconds) º 10 yards1 second15 dollars57. (8 hours) 45 dollars1 hour SOLVING EQUATIONS Solve the equation. Check your solution. (Review 1.3)3258. 3d 16 d º 459. 5 º x 23 2x60. 10(y º 1) y 461. p º 16 4 4(2 º p)62. º10x 5x 563. 12z 4z º 56264. x º 7 13365. º x 19 º114131166. x º x4855535167. x º x 4462Chapter 1 Equations and Inequalities
Jul 16, 2018 · Page 1 of 2 28 Chapter 1 Equations and Inequalities REWRITING COMMON FORMULAS Throughout this course you will be using many formulas. Several are listed below. Rewriting a Common Formula The formula for the perimeter of a rectangle is P 2 2w.Solve for w. SOLUTION P 2 2w Write perimeter formula. P º2 2w Subtract 2 from each side. P º 2
EQUATIONS AND INEQUALITIES Golden Rule of Equations: "What you do to one side, you do to the other side too" Linear Equations Quadratic Equations Simultaneous Linear Equations Word Problems Literal Equations Linear Inequalities 1 LINEAR EQUATIONS E.g. Solve the following equations: (a) (b) 2s 3 11 4 2 8 2 11 3 s
1 Solving Linear Equations 1.1 Solving Simple Equations 1.2 Solving Multi-Step Equations 1.3 Solving Equations with Variables on Both Sides 1.4 Rewriting Equations and Formulas Mathematical Thinking: Mathematically proficient students can apply the mathematics they know to solve problems arising in ev
1.4 Rewriting Equations and Formulas 29 1. Complete this statement: A w is an example of a(n) ?. 2. Which of the following are equations with more than one variable? A. 2x 5 9 º 5x B. 4x 10y 62 C. x º 8 3y 7 3. Use the equation from Example 3. Describe how you would solve for A. Solve the equation for
1.2 First Order Equations 5 1.3 Direction Fields for First Order Equations 14 Chapter 2 First Order Equations 2.1 Linear First Order Equations 27 2.2 Separable Equations 39 2.3 Existence and Uniqueness of Solutions of Nonlinear Equations 48 2.5 Exact Equations 55 2.6 Integrating Factors 63 Chapter 3 Numerical Methods 3.1 Euler’s Method 74
Chapter 1 Introduction 1 1.1 ApplicationsLeading to Differential Equations 1.2 First Order Equations 5 1.3 Direction Fields for First Order Equations 16 Chapter 2 First Order Equations 30 2.1 Linear First Order Equations 30 2.2 Separable Equations 45 2.3 Existence and Uniqueness of Solutionsof Nonlinear Equations 55
point can be determined by solving a system of equations. A system of equations is a set of two or more equations. To ÒsolveÓ a system of equations means to find values for the variables in the equations, which make all the equations true at the same time. One way to solve a system of equations is by graphing.
COMPLEXITY HIERARCHIES AND HIGHER-ORDER CONS-FREE TERM REWRITING CYNTHIA KOP AND JAKOB GRUE SIMONSEN Department of Computer Science, Copenhagen University e-mail address: fkop,simonseng@di.ku.dk Abstract. Constructor rewriting systems are said to be cons-free if, roughly, constructor .
50 80 100 150 200 250 300 350 400 450 500 550 600 . (API 624/ ISO 15848), cryogenic valves (-196 C) and valves in exotic metallurgies. Valves in other sizes and ASME classes available on demand. 4 Compliance Standards Parameter Standard Design Gate Valves API 603, ASME B16.34 Globe Valves ASME B16.34 Check Valves ASME B16.34 Ends Face-to-face/ End-to-end Dimensions ASME B16.10 End Flange .
