Solving Equations, Solving an Equation | HP 33s Scientific instruction

7

Solving Equations

In chapter 6 you saw how you can use to find the value of the left–hand variable in an assignment–type equation. Well, you can use SOLVE to find the value of any variable in any type of equation.

For example, consider the equation

x2 – 3y = 10

If you know the value of y in this equation, then SOLVE can solve for the unknown x. If you know the value of x, then SOLVE can solve for the unknown y. This works for "word problems" just as well:

Markup ⋅ Cost = Price

If you know any two of these variables, then SOLVE can calculate the value of the third.

When the equation has only one variable, or when known values are supplied for all variables except one, then to solve for x is to find a root of the equation. A root of an equation occurs where an equality or assignment equation balances exactly, or where an expression equation equals zero. (This is equivalent to the value of the equation being zero.)

Solving an Equation

To solve an equation for an unknown variable:

1.Press H and display the desired equation. If necessary, type the equation as explained in chapter 6 under "Entering Equations into the Equation List."

2.Press then press the key for the unknown variable. For example, press X to solve for x. The equation then prompts for a value for every other variable in the equation.

3.For each prompt, enter the desired value:

Solving Equations 7–1

Image 111

HP 33s Scientific manual Solving Equations, Solving an Equation, Markup ⋅ Cost = Price

Contents

Hp 33s scientific calculator Printing History Contents Clearing the X-Register RPN The Automatic Memory Stack Storing Data into Variables How You Can Use Equations Summary of Equation Operations Fractions in Programs Accuracy of Integration Complex Stack Complex Operations ContentsSolving an Equation For More Information Programming 12-7 12-4Data Input and Output 12-4 Entering a Program 12-5 12-6 Branching GTO 13-4 13-113-2 13-3 15-12 Managing Calculator Memory ContentsPart 3. Appendixes and Reference Vector Operations 15-1 About ALG Resetting the Calculator Clearing MemoryRound-Off Error Contents Disabling Operations Neutral Operations Underflow How the Integral Is EvaluatedConditions That Prolong Calculation Time Page Basic Operation PartPage Turning the Calculator On and Off Adjusting Display ContrastGetting Started Important Preliminaries Highlights of the Keyboard and Display Shifted Keys2Getting Started Cursor Keys Alpha Keys Silver Paint Keys Backspacing and Clearing4Getting Started ¾ when number entry is complete Keyboard-entry modeAre described in Using Menus on page 1-7. If Equation-entry mode 6Getting Started Numeric Functions Using MenusHP 33s Menus Chapter Name Programming Instructions To use a menu function Other functions8Getting Started Description KeysDisplay Exiting Menus RPN and ALG Keys RPN mode ALG modeTo select RPN mode To select ALG mode First Line Second Line Display and Annunciators RAD or HP 33s Annunciators Meaning Chapter12Getting Started 2 3 §,¨ Keys Display Description Keying in NumbersMaking Numbers Negative Exponents of Ten Understanding Digit Entry Keying in Exponents of TenOther Exponent Functions 16Getting Started Doing ArithmeticRange of Numbers and Overflow Keys Display Key in the number. You dont need to press ‘ One-Number FunctionsTwo-Number Functions , \, , Qor T D, , \, ,Q Controlling the Display FormatPeriods and Commas in Numbers ‘3 z Number of Decimal Places Fixed-Decimal Format %Scientific Format ALL Format SHOWing Full 12-Digit Precision20Getting Started Engineering Format Fractions Entering FractionsKeysDisplayDescription 22Getting Started Messages Displaying FractionsDisplayDescription 24Getting Started Calculator MemoryChecking Available Memory Clearing All of Memory RPN The Automatic Memory Stack What the Stack IsRPN The Automatic Memory Stack X and Y-Registers are in the Display Clearing the X-Register2RPN The Automatic Memory Stack Reviewing the Stack ¶ Roll DownRµ Roll Up Arithmetic How the Stack Does It Exchanging the X- and Y-Registers in the Stack4RPN The Automatic Memory Stack Lost How Enter Works Filling the stack with a constant Using a Number Twice in a RowHow Clear x Works 6RPN The Automatic Memory Stack Last X Register Wrong One-Number Function Correcting Mistakes with Last, \, , Qor T 8RPN The Automatic Memory Stack 96.704 + Reusing Numbers with Last› Mistake 10RPN The Automatic Memory Stack Work from the Parentheses Out Chain Calculations in RPN mode 12RPN The Automatic Memory Stack 16.3805x5 = 181.0000 SolutionExercises Order of Calculation More Exercises 14RPN The Automatic Memory Stack‘14 ‘ ⋅ 0.8 ⎟ 12.5 − 0.73 = Page Storing Data into Variables Storing Data into Variables Example Storing Numbers Storing and Recalling Numbers2Storing Data into Variables To recall a copy of a number from a variable to the display Viewing a Variable without Recalling It Reviewing Variables in the VAR CatalogTo review the values at any or all non-zero variables 4Storing Data into Variables Clearing VariablesArithmetic with Stored Variables Storage Arithmetic Result 3 That is, x Recall Arithmetic Exchanging x with Any Variable LzE6Storing Data into Variables Variable Page Real-Number Functions Exponential and Logarithmic FunctionsReal-Number Functions 2Real-Number Functions Power FunctionsQuotient and Remainder of Division To Calculate Press Trigonometry Entering πTo Calculate Press Result 4Real-Number Functions Setting the Angular ModeTrigonometric Functions Option Description Annunciator Programming Note NzR ‘x Q Hyperbolic FunctionsPercentage Functions 6Real-Number Functions 16.12 ‘ 15.76 T . Physics Constants Const Menu Items Description Value8Real-Number Functions Conversion Functions Items Description ValueTo insert a constant Coordinate Conversions 10Real-Number FunctionsTo convert between rectangular and polar coordinates Example Conversion with Vectors Example Polar to Rectangular Conversion Example Converting Time Formats Time Conversions12Real-Number Functions To convert between decimal fractions and minutes-seconds To convert an angle between degrees and radians Angle ConversionsUnit Conversions To Convert Press Displayed Results Probability Probability FunctionsFactorial Gamma Keys Example Combinations of PeopleDisplay 16Real-Number Functions Parts of Numbers Names of Functions Page Entering Fractions Fractions Fractions in the Display Display Rules2Fractions Entered Value Internal Value Displayed Fraction Accuracy Indicators Changing the Fraction Display Longer Fractions4Fractions Choosing a Fraction Format Setting the Maximum Denominator 6Fractions Examples of Fraction DisplaysTo Get This Fraction Format Change These Flags Fraction How 2.77 Is Displayed Format Rounding Fractions Fraction Number Entered and Fraction Displayed Format ¼Denominator Fixed 8Fractions Fractions in Equations Fractions in Programs Page Example Calculating with an Equation Entering and Evaluating EquationsHow You Can Use Equations Entering and Evaluating Equations ZNz ZLL2Entering and Evaluating Equations Enters and leaves Equation mode Summary of Equation OperationsKey Operation To enter an equation Entering Equations into the Equation ListVariables in Equations 4Entering and Evaluating Equations Functions in Equations Numbers in Equations ZLC z Displaying and Selecting EquationsParentheses in Equations LR d To select an equation Editing and Clearing Equations——— To view a long equation 8Entering and Evaluating Equations Evaluating Equations Types of Equations To evaluate an equation Type of Equation Result for ‘10Entering and Evaluating Equations Fx ¼ Example Evaluating an Equation with Enter Using Enter for Evaluation Example Evaluating an Equation with XEQ Using XEQ for EvaluationResponding to Equation Prompts 12Entering and Evaluating Equations „ To display digits hidden by the prompt, press Syntax of EquationsOperator Precedence ‘5 g Order Operation Example Equations Meaning14Entering and Evaluating Equations Σxy Equation Functions Perimeter = a + b + h Operation Equation function16Entering and Evaluating Equations Example Perimeter of a Trapezoid Area = HLA ’.25 zLN zLD 2 zR NqLN `qONqLN `‘Cos π / n Example Area of a Polygon Example Checksum and Length of an Equation Syntax ErrorsVerifying Equations 18Entering and Evaluating Equations Markup ⋅ Cost = Price Solving EquationsSolving an Equation Solving Equations Example Solving the Equation of Linear Motion LD dLVZLT › 2Solving Equations Value Example Solving the Ideal Gas Law Equation HL P zZL T 4Solving Equations Understanding and Controlling Solve LV q 6Solving Equations Verifying the Result Choosing Initial Guesses for Solve Interrupting a Solve Calculation 8Solving Equations Example Using Guesses to Find a Root 40 H ZL H 10Solving Equations For More Information Page Integrating Equations Integrating Equations Integrating Equations ≥ FN J0 x = π1 ≥0π cos x sin t dt2Integrating Equations Value ≥ 0π f t 4Integrating Equations Example Sine Integral= ≥ t Accuracy of Integration Example Specifying Accuracy Specifying AccuracyInterpreting Accuracy 6Integrating Equations Example Changing the Accuracy Its uncertainty indicates 8Integrating Equations To enter a complex number Operations with Complex NumbersComplex Stack + iy Complex Operations 2Operations with Complex NumbersTo do an operation with one complex number To do an arithmetic operation with two complex numbers Functions for One Complex Number, z To Calculate Press ‘2 ‘ ‘4 2 5 ‘4Operations with Complex Numbers Using Complex Numbers in Polar Notation2 3 ‘ ‘2 Example Vector Addition 6Operations with Complex Numbers Operations with Complex Numbers Page Base Conversions and Arithmetic Base Conversions and ArithmeticBase Menu Menu label Description 10-2Base Conversions and Arithmetic Arithmetic in Bases 2, 8, Keys Display 10-4Base Conversions and Arithmetic Representation of NumbersNegative Numbers ˜˜ 7FFFFFFFF Range of Numbers 10-6Base Conversions and Arithmetic Windows for Long Binary Numbers Statistical Operations Entering Statistical DataStatistical Operations 11-2Statistical Operations Correcting Errors in Data EntryEntering One-Variable Data Entering Two-Variable Data Initial x, y Corrected x, y 400‘20 11-4Statistical Operations Statistical CalculationsMean Statistics Menus Key Description Number of Parts y Example Weighted Mean Two Variables· Price per Part 11-6Statistical Operations Sample Standard DeviationPopulation Standard Deviation Uº Linear Regression Menu Linear Regressionσº Keys DisplayDescription Grain Yield 11-8Statistical OperationsExample Curve Fitting Nitrogen Applied 70, y Limitations on Precision of Data Normalizing Close, Large Numbers Summation Values and the Statistics RegistersSummation Statistics 11-10Statistical Operations # ™ ´º¸/ ™ ´¸/ ™ ´º/ ™ ´¸/ ™ ´º/ ‡ Access to the Statistics RegistersStatistics Registers in Calculator Memory Example Viewing the Statistics Registers 11-12Statistical Operations Programming Page Example a Simple Program RPN mode 5 !Nz ALG mode 5 !zN‘Simple Programming Simple Programming 12-2Simple Programming RPN mode ALG modeRPN mode Display Description Program Labels Selecting a ModeDesigning a Program Program Boundaries LBL and RTN ALG Operations Using RPN, ALG and Equations in ProgramsData Input and Output Strengths of RPN Operations To enter a program into memory Entering a Program Keys That Clear 12-6Simple ProgrammingTo enter an equation in a program line RPN mode Function Names in ProgramsExample Entering a Labeled Program Example Entering a Program with an Equation HN ZLR 12-8Simple Programming NzX Testing a ProgramRunning a Program Executing a Program XEQ 12-10Simple Programming Example Testing a Program Using Input for Entering Data Entering and Displaying Data@ To use the Input function in a program 12-12Simple Programming To respond to a prompt Using View for Displaying Data Using Equations to Display Messages 12-14Simple ProgrammingExample INPUT, VIEW, and Messages in a Program = 2π R2 + 2π RH = 2π R R + H NzLR LR z › LH ›p 12-16Simple Programming Displaying Information without Stopping Interrupting a Running Program Error StopsStopping or Interrupting a Program Programming a Stop or Pause STOP, PSE To insert a program line Editing a Program12-18Simple Programming To delete a program line Viewing Program Memory Program Memory 12-20Simple Programming Memory UsageCatalog of Programs MEM Clearing One or More Programs To clear all programs from memory Checksum V Selecting a Base Mode in a ProgramNonprogrammable Functions Programming with Base Ax4 + Bx3 + Cx2 + Dx + E Decimal mode set Hexadecimal mode setPolynomial Expressions and Horners Method Numbers Entered in Program Lines Keys Display ALG mode ‰X12-24Simple Programming Checksum and length E41A Page Programming Techniques Routines in ProgramsProgramming Techniques 13-2Programming Techniques Calling Subroutines XEQ, RTN Example a Nested Subroutine Nested Subroutines Branching GTO 13-4Programming TechniquesA2 + B2 Programmed GTO Instruction Using GTO from the Keyboard 13-6Programming Techniques Conditional Instructions Test Menus Tests of Comparison x?y, x?0 !- % ! ! ! º¸@ ! Program Lines Description RPN modeFlags 13-8Programming Techniques Flag Fraction-Control Flags Status ClearSet 13-10Programming Techniques Using Flags Flags Menu Menu Key DescriptionAnnunciators for Set Flags 13-12Programming Techniques Example Using Flags Program Lines Description RPN mode $ @ $ $ @ $ Example Controlling the Fraction Display 13-14Programming Techniques Program Lines Description 13-16Programming Techniques Loops Conditional Loops GTO ! ! . ! º6¸@ ! #$ ! Loops with Counters DSE, ISG 13-18Programming TechniquesLoop-Control Number $ $ ! $$ Indirectly Addressing Variables and Labels Variable13-20Programming Techniques If i contains Then i will address Indirect Address ! L Program Control with13-22Programming Techniques Example Choosing Subroutines With Example Loop Control With If i holds Then XEQi callsProgram lines - L & %1L2 13-24Programming Techniques Equations with L 1L2 Page To solve a programmed function Solving and Integrating ProgramsSolving a Program Solving and Integrating Programs Example Program Using ALG 14-2Solving and Integrating ProgramsV= N x R x T Pressure ⋅ volume. Pressure ⋅ volume Example Program Using Equation LP z LV ’ LN z LR z LT ‘14-4Solving and Integrating Programs @ @ ‘10 !@ # / Example Solve in a Program Using Solve in a Program14-6Solving and Integrating Programs / label #variable Program Lines In RPN mode Integrating a ProgramTo integrate a programmed function To write a program for ≥ FN 14-8Solving and Integrating Programs Using Integration in a Program Sin/ label 14-10Solving and Integrating Programs ≥ G variableExample ≥ FN in a Program Restrictions on Solving and Integrating Page = R cosP = arctan Mathematics ProgramsVector Operations Mathematics Programs 15-2Mathematics Programs = XU + YV + ZWV1 = X i + Y j + Z k V2=U i + V j + W k Program Listing X2 + Y2 + Z2 and P 15-4Mathematics Programs Program Lines Description ALG mode Goes to the vector-addition routine 15-6Mathematics Programs Flags Used RemarksProgram Instructions Variables Used 15-8Mathematics Programs XR 15-10Mathematics Programs@ 07 g !@ 125 g @ Solutions of Simultaneous Equations 15-12Mathematics ProgramsAX + DY + GZ = J BX + EY + HZ = K CX + FY + IZ = L % 15-14Mathematics Programs Mathematics Programs 15-16Mathematics Programs Flags Used Scratch 15-18Mathematics Programs @ @ @ Polynomial Root Finder15-20Mathematics Programs @ ⋅ the sign of JK a1/2 Where J = a3/2 = y0 /2X2 + J + Lx + K + M = X2 + J Lx + K M = 15-22Mathematics Programs Polynomial ! 15-24Mathematics Programs B1 = a3a1 4a0 Checksum and length C7A6B0 =a04a0 a32 a12 B2= -a2 JK. a1 15-26Mathematics Programs + L + M 15-28Mathematics Programs Order Terms and Coefficients 15-30Mathematics Programs %/ 15-32Mathematics Programs Coordinate Transformations Old coordinate System 15-34Mathematics Programs Checksum and length 8C82 Remark 15-36Mathematics Programs Ÿ @ 15-38Mathematics Programs Statistics Programs Curve FittingStatistics Programs 16-2Statistics Programs = B + MIn Description 16-4Statistics Programs #$ #$ 16-6Statistics Programs 16-8Statistics Programs %@ 37.9 g &@ XS%@ 40.5 g &@ value 104.5 g %@ 38.6 g &@ 16-10Statistics Programs // / %@ Normal and Inverse-Normal Distributions Logarithmic Exponential Power 16-12Statistics Programs Function ª º ª -+. H % ! 16-14Statistics Programs Variable used temporarily to pass the value S ⋅ 2 π to Inverse programXS@ 16-16Statistics Programs 2f − Grouped Standard Deviation 16-18Statistics Programs Updates ƒxi 2fi in register Increments or decrements N 16-20Statistics Programs XU/ %@ 13 g @ / Group/ %@ @ %@ 16-22Statistics Programs%@ %@ Miscellaneous Programs and Equations Time Value of MoneyMiscellaneous Programs and Equations 17-2Miscellaneous Programs and Equations …LN `QLI ›LF z ›LI Solve instructions Keys Display Description RPN mode šas17-4Miscellaneous Programs and Equations Retains 5750 in B calculates monthly interest rate 17-6Miscellaneous Programs and Equations Prime Number Generator LBL Y 17-8Miscellaneous Programs and Equations #$ No test is made to ensure that the input is greater than 17-10Miscellaneous Programs and Equations Appendixes and Reference Page Answers to Common Questions Support, Batteries ServiceSupport, Batteries, and Service A-1 Calculator Support 2Support, Batteries, and Service Environmental LimitsChanging the Batteries To install batteries Support, Batteries, and Service A-3 Fire. The batteries can burst or explode Testing Calculator Operation4Support, Batteries, and Service Do not mutilate, puncture, or dispose of batteries Support, Batteries, and Service A-5 Self-Test™Ÿ˜—š 6Support, Batteries, and Service Warranty Service Support, Batteries, and Service A-7Europe Country Telephone numbers 8Support, Batteries, and Service Asia Pacific Country Telephone numbersAmerica Country Telephone numbers Support, Batteries, and Service A-9 Regulatory InformationCanada Japan 10Support, Batteries, and Service User Memory and the Stack Managing Calculator MemoryUser Memory and the Stack 2User Memory and the Stack Resetting the Calculator Default Clearing MemoryCategory 4User Memory and the Stack Status of Stack LiftDisabling Operations Neutral Operations EQN FDISP Vlabel nnnn Status of the Last X Register 6User Memory and the Stack€ θ ,r ALG Summary About ALGALG Summary C-1 2ALG Summary Power FunctionsDoing Two-number Arithmetic in ALG Simple Arithmetic Percentage Calculations ‘27 Q…ALG Summary C-3 4ALG Summary Permutations and CombinationsQuotient and Remainder Of Division 16.12 T ALG Summary C-5 Keys Display Description 6ALG Summary Reviewing the Stack ALG Summary C-7 Coordinate Conversions Integrating an Equation Q3 z8ALG Summary Complex Operations Operations with Complex NumbersALG Summary C-9 To view the result of complex operations 2 ›3 ` ` Q2 › 10ALG Summary ºy4 ÃË2 Ë5 ¹cº  ºy3 ÃË ALG Summary C-11 12ALG Summary Initial x, y Corrected x, y ALG Summary C-13 20 Page More about Solving How Solve Finds a RootMore about Solving D-1 Function Whose Roots Can Be Found 2More about Solving Interpreting Results More about Solving D-3Example An Equation With One Root Example An Equation with Two Roots 4More about Solving -%. More about Solving D-5/ / Example Discontinuous Function LX `’1.5 6More about Solving More about Solving D-7 − 1= 8More about Solving When Solve Cannot Find a Root Example a Relative Minimum More about Solving D-9 Example An Asymptote 10More about Solving Example Find the root of the equation ⎟ x + 0.3 − 0.5 =#LX q LX ›3 ``… More about Solving D-11 12More about Solving !1%ª1%-22Example a Local Flat Region More about Solving D-13 Round-Off Error 14More about Solving Underflow More about Integration How the Integral Is EvaluatedMore about Integration 2More about Integration Conditions That Could Cause Incorrect Results ≥0∞ xe −x dx 4More about Integration LX zLX ` ‡ More about Integration 6More about Integration Conditions That Prolong Calculation Time . 8More about Integration More about Integration Page Messages F-1 Messages Or program. This might take a while 2Messages # variable Messages F-3 Self-Test Messages 4Messages Operation Index Name Keys and DescriptionOperation Index G-1 2Operation Index ˜or —j h Polar to rectangular coordinates Operation Index G-3 4Operation Index P Operation Index G-5 M …Complex Subtraction 6Operation IndexGComplex Change sign ›Complex Operation Index G-7 DSE variable Mvariable 8Operation IndexRHyperbolic VRadians to degrees ?Fractional part Operation Index G-9Natural exponential Wlabel Nnnn 10Operation IndexGTO label Vlabel label Integer part ISG variable Lvariable ”I”Operation Index G-11 Input variable ‰variable Common logarithm 12Operation IndexLBL label •label Natural logarithm WDegrees to radians Operation Index G-13Permutations of n items FPause 14Operation Index Operation Index G-15 16Operation Index Operation Index G-17 18Operation Index 11-10 Given an x-value Operation Index G-19≠0? 13-7 If x=0, skips the next program Line ≤0? 13-7 20Operation Index Power Returns y raised to the xth power Index-1 Index Index-2 Index-3 Index-4 Index-5 Index-6 Index-7 Index-8 Index-9 Index-10 Index-11 RPN Index-12 Index-13 View Index-14 Index-15