1.4 Rewriting Equations And Formulas
English1.4SpanishRewriting Equations and FormulasHow can you use a formula for onemeasurement to write a formula for a different measurement?1ACTIVITY: Using Perimeter and Area FormulasWork with a partner.a. Write a formula for the perimeter P ofa rectangle. Solve the formula for w. Use the new formula to find the widthof the rectangle.b.A 10 in.2hWrite a formula for the area A ofa triangle. Solve the formula for h. Use the new formula to find theheight of the triangle. Write a formula for the circumference Cof a circle. Solve the formula for r. Use the new formula to find the radius ofthe circle.b 4 in.A 15 in.2C 8π cmrrd.h 5.5 in. 5b in.2c.P 19 in.w Write a formula for the area A ofa trapezoid. Solve the formula for h. Use the new formula to find theheight of the trapezoid.B 6 in.e. Write a formula for the area A ofa parallelogram. Solve the formula for h. Use the new formula to find the heightof the parallelogram.A 56 m2hb 8m24Chapter 1Solving Equations
EnglishSpanish2ACTIVITY: Using Volume FormulasWork with a partner.a. Write a formula for the volume Vof a prism. Solve the formula for h. Use the new formula to find theheight of the prism.b.hB 12 in.2 Write a formula for the volume Vof a pyramid. Solve the formula for B. Use the new formula to find thearea of the base of the pyramid.V 48 ft3h 9 ftV 60 in.3Bc. Write a formula for the volume Vof a cylinder. Solve the formula for B. Use the new formula to find thearea of the base of the cylinder.V 48π cm3h 12 cmBd.V 18π m3hB 9π m2 Write a formula for the volume Vof a cone. Solve the formula for h. Use the new formula to find theheight of the cone.3. IN YOUR OWN WORDS How can you use a formula for one measurementto write a formula for a different measurement? Give an example that isdifferent from the examples on these two pages.Use what you learned about rewriting equations and formulas tocomplete Exercises 3 and 4 on page 28.Section 1.4Rewriting Equations and Formulas25
English1.4SpanishLessonLesson TutorialsAn equation that has two or more variables is called a literal equation.To rewrite a literal equation, solve for one variable in terms of the othervariable(s).Key Vocabularyliteral equation, p. 26EXAMPLE1Rewriting an EquationSolve the equation 2y 5x 6 for y.2y 5x 6Undo the addition.Write the equation.2y 5x 5x 6 5xSubtract 5x from each side.2y 6 5x2y2Simplify.6 5x2— —Undo the multiplication.Divide each side by 2.52y 3 —xSimplify.Solve the equation for y.1. 5y x 10Exercises 5 –10EXAMPLE22.4x 4y 13.12 6x 3yRewriting a FormulaThe formula for the surface area S of a cone is S 𝛑 r 2 𝛑 rℓ. Solvethe formula for the slant height ℓ.RememberA formula shows howone variable is relatedto one or more othervariables. A formula is atype of literal equation.S π r 2 π rℓWrite the equation.S π r 2 π r 2 π r 2 π rℓSubtract π r 2 from each side.S π r 2 π rℓSimplify.S π r 2 π rℓ— —Divide each side by π r.S πrπrSimplify.πrπrr2— ℓSolve the formula for the red variable.Exercises 14 – 194. Area of rectangle: A bh5.Simple interest: I Prt6. Surface area of cylinder: S 2π r 2 2π r h26Chapter 1Solving Equations
EnglishSpanishTemperature ConversionA formula for converting from degrees Fahrenheit F to degreesCelsius C is59C — (F 32).EXAMPLERewriting the Temperature Formula3Solve the temperature formula for F.59C — (F 32)95—Use the reciprocal.Write the temperature formula. C —95 —59 (F 32)Multiply each side by —, the reciprocal of —.95Simplify.95— C F 32Undo the subtraction.— C 32 F 32 3295Add 32 to each side.95Simplify.— C 32 F5995The rewritten formula is F — C 32.EXAMPLE4Real-Life ApplicationWhich has the greater temperature?Sun11,000 FConvert the Celsius temperature of lightning to Fahrenheit.95Write the rewritten formula from Example 3. — (30,000) 3295Substitute 30,000 for C. 54,032Simplify.F — C 32Lightning30,000 CBecause 54,032 F is greater than 11,000 F, lightning has thegreater temperature.7. Lake Kissimmee has a water temperature of 85 F.Lake Okeechobee has a water temperature of 30 C.Which lake has the greater water temperature? Explain.Section 1.4Rewriting Equations and Formulas27
EnglishSpanishExercises1.4Help with Homework381. VOCABULARY Is 2x — a literal equation? Explain.2. DIFFERENT WORDS, SAME QUESTION Which is different? Find “both” answers.Solve 4x 2y 6 for y.Solve 6 4x 2y for y.Solve 4x 2y 6 for y in terms of x.Solve 4x 2y 6 for x in terms of y.6) 39 (- 3) 3 (- 9) 4 (- 1)9 (-3. a. Write a formula for the area Aof a triangle.4.a. Write a formula for the volume Vof a prism.b. Solve the formula for b.b. Solve the formula for B.c. Use the new formula to find thebase of the triangle.c.A 36 mm2Use the new formula to find thearea of the base of the prism.V 36 in.3h 6 in.h 6 mmBbSolve the equation for y.113155. —x y 46. 3x — y 78. π 7x 2y9. 4.2x 1.4y 2.111. ERROR ANALYSIS Describe and correct theerror in rewriting the equation.7. 6 4x 9y 10. 6y 1.5x 82x y 5y 2x 512. TEMPERATURE The formula K C 273.15converts temperatures from Celsius C to Kelvin K.a. Solve the formula for C.b. Convert 300 K to Celsius.I 75P 500a. Solve the formula for t.r5%b. Use the new formula to find the value of t in the table.t13. INTEREST The formula for simple interest is I Prt.28Chapter 1Solving Equations
EnglishSpanishSolve the equation for the red variable.15. e mc 22 14. d rt1216. R C PVh17. A — π w 2 2ℓw1618. B 3 —19. g —(w 40)20. WRITING Why is it useful to rewrite a formula in termsof another variable?5921. TEMPERATURE The formula K — (F 32) 273.15converts temperatures from Fahrenheit F to Kelvin K.a. Solve the formula for F.b. The freezing point of water is 273.15 Kelvin.What is this temperature in Fahrenheit?c. The temperature of dry ice is 78.5 C.Which is colder, dry ice or liquid nitrogen?77.35 KNavy Pier Ferris Wheel222.FERRIS WHEEL The Navy Pier Ferris Wheel in Chicagohas a circumference that is 56% of the circumferenceof the first Ferris wheel built in 1893.C 439.6 fta. What is the radius of the Navy Pier Ferris Wheel?b. What was the radius of the first Ferris wheel?c. The first Ferris wheel took 9 minutes to makea complete revolution. How fast was thewheel moving?23.The formula for the volume of a sphere is43V —π r 3. Solve the formula for r 3. Use guess, check,and revise to find the radius of the sphere.V â 381.51 in.3Multiply.rSKILLS REVIEW HANDBOOK3424. 5 —8325. 2.4 —1432893526. — — —11227. 25 — —328. MULTIPLE CHOICE Which of the following is not equivalent to — ?4 SKILLS REVIEW HANDBOOKA 0.75 B 3:4 C 75% Section 1.4D 4:3 Rewriting Equations and Formulas29
Section 1.4 Rewriting Equations and Formulas 25 Work with a partner. a. Write a formula for the volume V of a prism. Solve the formula for h. Use the new formula to fi nd the height of the prism. b. Write a formula for the volume V of a pyramid. Solve the formula for B. Use the new formula to fi nd the area of the
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 .
