Chapter 4: Solving Literal Equations

Chapter 4:Solving LiteralEquations1

Day 1: The Basics of Literal EquationsA-REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.Warm-UpWhat is a literal equation?A literal equation is an equation with two or more variables. Instead of solving for a numerical value, we solvefor one variable in terms of another. A formula is one type of literal equation that has special applications inmath or science.Observe the similarities between the linear equation (left) and the literal equation (right):One-Step Linear Equation:1) y 10 553)s 40 85 solve for sTwo-Step Linear Equation:5) 2a 13 87 solve for aOne-Step Literal Equation:2)y x 55 solve for y4)s x 85 solve for sTwo-Step Literal Equation:6)2a b c solve for a2

Solving for a variable using division:7)3x 458)3x y solve for xQuick Check for Understanding9)2 x y 9 solve for y10) 3b c d solve for b11)P mvsolve for mApplication12) The formula d rt relates the distance an object travels, d, to its average rate of speed r, and amount oftime t that it travels.a) Solve the formulad rtfor t.b) How many hours would it take for a car to travel 150 miles at an average rate of 50 miles per hour?3

Independent PracticeSolve for the variable indicated.1) 𝑑 𝑟𝑡 𝑆𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑟2) 𝑃 𝑎 𝑏 𝑆𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑏3) 𝑦 𝑚𝑥 𝑏 𝑆𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑥4) 𝑇 𝑀 𝑁 𝑆𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑁5) 𝐴𝑥 𝐵 𝐶 𝑆𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑥6) 𝐴𝑥 𝐵𝑦 𝐶 𝑆𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑦7) 𝐼 𝑝𝑟𝑡 𝑆𝑜𝑙𝑣𝑒𝑓𝑜𝑟 𝑟8) 𝐶 𝜋𝑑 𝑆𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑑4

9) 𝑠 𝑟𝜃 𝑆𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑟10) 𝐸 𝐼𝑅 𝑆𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑅11) 𝐸 𝑚𝑐 2 𝑆𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑐 212) 𝑃 13) 5𝑚 𝑛 10 𝑆𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑚14) 5 𝑏 2𝑡 𝑆𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑡15) 𝑃𝑉 𝑛𝑅𝑇 𝑆𝑜𝑙𝑣𝑒𝑓𝑜𝑟 𝑅16) 𝑦 2𝑙 2𝑤𝑆𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑦3𝑥 1 𝑆𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑥5

17) The volume of a prism is 𝑉 𝑙𝑤ℎ.a) Solve this formula for ℎ.b) If the volume of a prism is 64, its length 4, and its width 2, what is its height?SummaryHomeworkChapter 4- Day 1 -Textbook pp. 109-110 #2, 5, 8, 9, 10, 11, 14, 20, 23, 26, 36-376

Day 2: Solving Literal Equations with ProportionsA-REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.Warm-UpModel ProblemsSolving ProportionsSolve for x in each equation.Linear Equations:𝑥1) 933) 2𝑥 1 32Literal Equation:𝑥2) 𝑦34)𝑥2 𝑚 67

25) 10 (x 4)36) D 115(𝑥 15)Reminder: Don’t distribute acoefficient unless absolutely necessary!Application57) The formula to convert Celsius to Fahrenheit is given by C (𝐹 32) .9a) Solve this formula for F.b) The boiling point of water is 100 . What is the Fahrenheit equivalent of this temperature?8

8) Check for Understandinga)𝑑 𝑐𝑛Solve for the given variable.b)solve for n𝐴 𝑎 𝑏2c)𝑠𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑏𝐹 𝐺𝑚1 𝑚2𝑟2solve for 𝑚11d) The formula for the mean (average) 𝐴 of two numbers y and z is one-half their sum, or 𝐴 (𝑦 𝑧).2If the average of two numbers is 7 and one of the numbers is 4, find the other number.Cumulative Independent Practice1)𝑚𝑘 𝑥 𝑓𝑜𝑟 𝑘Days 1-2Solve for the value of the variable.12) 𝑉 𝐴ℎ 𝑓𝑜𝑟 𝐴39

13) 𝑠 𝑔𝑡 2 𝑓𝑜𝑟 𝑔25) 𝑞 𝑟 7)𝑥7𝑝5𝑓𝑜𝑟 𝑞 𝑦 𝑡 𝑓𝑜𝑟 𝑥4) 𝑠 𝑤 10𝑒𝑚6) 𝑞 𝑟 8)𝑥 𝑦7𝑝5𝑓𝑜𝑟 𝑤𝑓𝑜𝑟 𝑝 𝑡 𝑓𝑜𝑟 𝑥10

9)𝑥 𝑦7 𝑡 𝑓𝑜𝑟 𝑦11)𝑅 𝐶 𝑆13)𝑚 𝑦2 𝑦1𝑡𝑓𝑜𝑟 𝐶𝑥2 𝑥1𝑓𝑜𝑟 𝑦210)𝑃 𝑅 𝐶 𝑓𝑜𝑟 𝐶12)2𝑥 7𝑦 14 𝑓𝑜𝑟 𝑦14)𝑉 (𝑥 2𝑦) for x23115) The formula 𝑉 𝜋𝑟 2ℎ is the formula for the volume of a cylinder. To the nearest tenth, what is the height3of a cylinder with volume 100 cm3 and radius 2 cm?HW Chapter 4-Day 2 Textbook pp. 109-110 #6, 7, 12, 13, 15, 18, 24, 30, 4511

Day 3: Using the Distributive Property and Rational EquationsA-REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.Warm-UpThe formula 𝑃 2(𝐿 𝑊) is the formula for the perimeter of a rectangle. Solve this formula for L.What is the length of a rectangle whose perimeter is 48 and whose width is 6?Distribution and Reverse distribution1) When there is a common factor in all terms of an expression, we can use the distributive property in reverseto write it in factored form.Simplest forma)2𝐿 2𝑊Factored Form2(𝐿 𝑊)b) 3𝑎 3𝑏c)2𝑙𝑤 2𝑙d) 𝑓𝑏 𝑓𝑎e) 2𝜋𝑟ℎ 2𝜋𝑟 22) Model ProblemUsing the Distributive Property in ReverseSolve for c in terms of a and b: 𝑎𝑐 𝑏𝑐 𝑎𝑏12

3) Practicea)b)Using Rational Equations4) Model Problem11 1relates an object’s distance, 𝑎, and its image’s distance, 𝑏, to the focal length of the𝑎𝑏𝑓lens, 𝑓. Solve this formula for 𝑓.The formula5) PracticeThe total resistance in a circuit is given by the formula1𝑅 1𝑅1 1. Solve this formula for 𝑅1.𝑅213

Unit Summary So Far:Look for STRUCTURE in equations:One- or Two-Step EquationsProportions𝐴𝑥 𝐵 𝐶𝑥 𝑏 𝑐Reverse Distribution (Common Factor)𝐷 𝑀𝐾 𝑉1) 𝑟 𝑝𝑛 for n3) 𝑠 2𝑥 𝑡𝑟𝑓𝑜𝑟 𝑥2𝑚𝑣 2𝑥̅ 𝑥1 𝑥22Rational Equations (Sums and Differences)111 𝐶 𝐶1 𝐶2S 2𝜋𝑟 2 2𝜋𝑟ℎCumulative Practice/Homework Chapter 4 – Day 31Solve for the requested variable.2) 𝑉 13𝐵ℎ 𝑓𝑜𝑟 𝐵4) 𝑣 𝑣0 𝑎𝑡 for 𝑣014

5) 𝐽 𝑚𝑣𝑓 𝑚𝑣𝑖 for m6) 𝐸 𝐼𝑅 𝑓𝑜𝑟 𝐼7) 𝑦 𝑦1 𝑚(𝑥 𝑥1 )𝑓𝑜𝑟 𝑥8) 𝑈 𝑚𝑔ℎ 𝑓𝑜𝑟 𝑔9) 𝑅 𝜌𝑙𝐴𝑓𝑜𝑟 𝐴10) 𝐹 𝑉 𝐸 2 𝑓𝑜𝑟 𝑉15

11) 𝑈 12𝑄𝑉 𝑓𝑜𝑟 𝑉13) 𝐺 𝐻 𝑇𝑆 𝑓𝑜𝑟 𝐻15) 𝑃 𝑃0 𝜌𝑔ℎ for h12) 𝑧 𝑦 𝑥 𝑥𝑦 2 𝑓𝑜𝑟 𝑥14) 𝐹𝐶 16) 𝑒 𝑚𝑣 2𝑟𝑇𝐻 𝑇𝐶𝑇𝐻for m𝑓𝑜𝑟 𝑇𝐻16

Multiple Choice Practice.17)18)Real-World Application.19)If Patty paid 0.93 to mail her letter, how manyounces was it? (C cost, z ounces)17

Day 4: Square RootsA-REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.Warm-UpMini-Lesson: Using Square RootsTo solve for a squared variable, take its square root.Linear Equation:64 16𝑥 2Check for UnderstandingLiteral Equation:𝐴 𝜋𝑟 2 solve for rSolve for the indicated variable.11) The formula for kinetic energy is 𝐾 𝑚𝑣 2.2Write an expression for 𝑣 in terms of K and 𝑚.2) The gravitational force F that two planetary bodiesexert on one another is given by 𝐹 Solve this formula for 𝑟.𝐺𝑚1 𝑚2𝑟2.18

Cumulative Practice/Homework1)𝑉 𝑙𝑤ℎSolve for h3) A 1 h(b1 b2 )Solve for h25) Solve 𝑅 Solve for the value of the indicated variable.𝑙 3𝑤2for w12) 𝑠 𝑎𝑡 2 solve for t24) 𝑆𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑟:𝑝 𝑟3 𝑚 56) Solve 𝑎𝑥 𝑏𝑦 𝑐 0 𝑓𝑜𝑟 𝑦19

7) 𝐴 2𝜋𝑟ℎ 2𝜋𝑟 2 Solve for π8) Rewrite 𝐾 9) 𝑞 3𝑟 210) In 𝑎 𝑎𝑥 𝑏 , what is a in terms of x and b?11)5 𝑐6𝑆𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑟 𝑑 7 Solve for c12) 𝑎𝑐 𝑣2𝑟32𝑘𝑇 solved for T in terms of k and T.Solve for 𝑣20

Regents Practice.13)14)15)16)21

Day 5: ReviewA-REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.Look for STRUCTURE in equations:One-Step Equations1) 𝐼 𝑝𝑟𝑡 Solve for 𝑟.2) 𝑇 𝑀 𝑁 Solve for M.Two-Step Equations3) 5𝑡 2𝑟 25 Solve for t.4) 𝑣𝑡 16𝑡 2 Solve for v.Proportions5) 𝐹 𝑙𝑡7) 𝐴 1𝑑2Solve for l.ℎ(𝑎 𝑏) Solve for a.6) 𝑃 8) 𝑚 144𝑝𝑦Solve for p.𝑦2 𝑦1𝑥2 𝑥1Solve for 𝑦222

Reverse Distribution9) 𝑆 𝑅 𝑟𝑅Solve for R.10) 𝑎𝑥 𝑏𝑥 𝑐Solve for x.Rational Equations11)1𝐶 1𝐶1 1𝐶2𝑥𝑥34Solve for 𝐶112) Solve for v.14) 𝑉 𝑑Solve for x.Square Roots13) 𝐾 12𝑚𝑣 213𝜋𝑟 2ℎ Solve for r.Applications. The surface area of a sphere is given by the formula 𝑆 4𝜋𝑟 2. Solve this formula for r. Whatis the radius of a sphere whose surface area is 201 𝑐𝑚2?23

A-REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Look for STRUCTURE in equations: One-Step Equations 1) Solve for . 2) Solve for M. Two-Step Equations 3) 5 2 25