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HP 35s HP 35s scientific calculator - User Guide - Page 123

Solving Equations

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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. Solving an Equation To solve an equation (excluding built-in equations) for an unknown variable: 1. Press  and display the desired equation. If necessary, type the equation as explained in chapter 6 under "Entering Equations into the Equation List." Solving Equations 7-1

Section	Page
Contents	3
Part 1. Basic Operation	3
Part 2. Programming	9
Part 3. Appendixes and Reference	11
Getting Started	17
Important Preliminaries	17
Turning the Calculator On and Off	17
Adjusting Display Contrast	17
Highlights of the Keyboard and Display	18
Shifted Keys	18
Alpha Keys	19
Cursor Keys	19
Backspacing and Clearing	20
Using Menus	22
Exiting Menus	24
RPN and ALG Modes	25
Undo key	27
The Display and Annunciators	28
Keying in Numbers	31
Making Numbers Negative	31
Exponents of Ten	31
Understanding Entry Cursor	33
Range of Numbers and OVERFLOW	33
Performing Arithmetic Calculations	34
Single Argument or Unary Operations	34
Two Argument or Binary Operations	35
Controlling the Display Format	37
Periods and Commas in Numbers (') (,)	39
Complex number display format (xy, x+y, r·‚)	40
SHOWing Full 12-Digit Precision	41
Fractions	42
Entering Fractions	42
Messages	43
Calculator Memory	44
Checking Available Memory	44
Clearing All of Memory	45
RPN: The Automatic Memory Stack	47
What the Stack Is	47
The X and Y-Registers are in the Display	49
Clearing the X-Register	49
Reviewing the Stack	49
Exchanging the X- and Y-Registers in the Stack	50
Arithmetic - How the Stack Does It	51
How ENTER Works	52
How to Clear the Stack	53
The LAST X Register	54
Correcting Mistakes with LAST X	55
Reusing Numbers with LAST X	56
Chain Calculations in RPN Mode	58
Work from the Parentheses Out	58
Exercises	60
Order of Calculation	60
More Exercises	62
Storing Data into Variables	65
Storing and Recalling Numbers	66
Viewing a Variable	68
Using the MEM Catalog	68
The VAR catalog	68
Arithmetic with Stored Variables	70
Storage Arithmetic	70
Recall Arithmetic	71
Exchanging x with Any Variable	72
The Variables \	73
Real-Number Functions	75
Exponential and Logarithmic Functions	75
Quotient and Remainder of Division	76
Power Functions	76
Trigonometry	77
Entering p	77
Setting the Angular Mode	78
Trigonometric Functions	78
Hyperbolic Functions	80
Percentage Functions	80
Physics Constants	82
Conversion Functions	84
Rectangular/Polar Conversions	84
Time Conversions	87
Angle Conversions	87
Unit Conversions	88
Probability Functions	89
Factorial	89
Gamma	89
Probability	89
Parts of Numbers	91
Fractions	93
Entering Fractions	93
Fractions in the Display	94
Display Rules	94
Accuracy Indicators	95
Changing the Fraction Display	96
Setting the Maximum Denominator	96
Choosing a Fraction Format	98
Examples of Fraction Displays	100
Rounding Fractions	100
Fractions in Equations	101
Fractions in Programs	102
Entering and Evaluating Equations	103
How You Can Use Equations	103
Summary of Equation Operations	105
Entering Equations into the Equation List	106
Variables in Equations	106
Numbers in Equations	107
Functions in Equations	107
Parentheses in Equations	108
Displaying and Selecting Equations	108
Editing and Clearing Equations	110
Types of Equations	111
Evaluating Equations	112
Using ENTER for Evaluation	113
Using XEQ for Evaluation	114
Responding to Equation Prompts	115
The Syntax of Equations	116
Operator Precedence	116
Equation Functions	118
Syntax Errors	121
Verifying Equations	121
Solving Equations	123
Solving an Equation	123
Solving built-in Equation	128
Understanding and Controlling SOLVE	129
Verifying the Result	129
Interrupting a SOLVE Calculation	130
Choosing Initial Guesses for SOLVE	130
For More Information	134
Integrating Equations	135
Integrating Equations ( Ú FN)	136
Accuracy of Integration	140
Specifying Accuracy	140
Interpreting Accuracy	140
For More Information	142
Operations with Complex Numbers	143
The Complex Stack	144
Complex Operations	144
Using Complex Numbers in Polar Notation	147
Complex Numbers in Equations	149
Complex Number in Program	150
Vector Arithmetic	151
Vector operations	151
Absolute value of the vector	153
Dot product	154
Angle between vectors	155
Vectors in Equations	156
Vectors in Programs	157
Creating Vectors from Variables or Registers	158
Base Conversions and Arithmetic and Logic	159
Arithmetic in Bases 2, 8, and 16	162
The Representation of Numbers	164
Negative Numbers	164
Range of Numbers	165
Windows for Long Binary Numbers	166
Using base in program and equations	166
Statistical Operations	167
Entering Statistical Data	167
Entering One-Variable Data	168
Entering Two-Variable Data	168
Correcting Errors in Data Entry	168
Statistical Calculations	170
Mean	170
Sample Standard Deviation	172
Population Standard Deviation	173
Linear Regression	173
Limitations on Precision of Data	176
Summation Values and the Statistics Registers	177
Summation Statistics	177
Access to the Statistics Registers	178
Simple Programming	183
Designing a Program	185
Selecting a Mode	185
Program Boundaries (LBL and RTN)	186
Using RPN, ALG and Equations in Programs	186
Data Input and Output	187
Entering a Program	188
Clear functions and backspace key	189
Function Names in Programs	190
Running a Program	192
Executing a Program (XEQ)	192
Testing a Program	193
Entering and Displaying Data	194
Using INPUT for Entering Data	195
Using VIEW for Displaying Data	197
Using Equations to Display Messages	198
Displaying Information without Stopping	200
Stopping or Interrupting a Program	201
Programming a Stop or Pause (STOP, PSE)	201
Interrupting a Running Program	201
Error Stops	201
Editing a Program	202
Program Memory	203
Viewing Program Memory	203
Memory Usage	204
The Catalog of Programs (MEM)	204
Clearing One or More Programs	205
The Checksum	205
Nonprogrammable Functions	206
Programming with BASE	206
Selecting a Base Mode in a Program	207
Numbers Entered in Program Lines	207
Polynomial Expressions and Horner's Method	208
Programming Techniques	211
Routines in Programs	211
Calling Subroutines (XEQ, RTN)	211
Nested Subroutines	212
Branching (GTO)	214
A Programmed GTO Instruction	215
Using GTO from the Keyboard	215
Conditional Instructions	216
Tests of Comparison (x?y, x?0)	217
Flags	219
Loops	226
Conditional Loops (GTO)	227
Loops with Counters (DSE, ISG)	228
Indirectly Addressing Variables and Labels	230
The Variables \	230
The Indirect Address, (I) and (J)	231
Program Control with (I)/(J)	233
Equations with (I)/(J)	233
Unnamed indirect variables	233
Solving and Integrating Programs	235
Solving a Program	235
Using SOLVE in a Program	240
Integrating a Program	241
Using Integration in a Program	244
Restrictions on Solving and Integrating	245
Statistics Programs	247
Curve Fitting	247
Normal and Inverse-Normal Distributions	257
Grouped Standard Deviation	264
Miscellaneous Programs and Equations	271
Time Value of Money	271
Prime Number Generator	277
Cross Product in Vectors	281
Support, Batteries, and Service	287
Calculator Support	287
Answers to Common Questions	287
Environmental Limits	288
Changing the Batteries	289
Testing Calculator Operation	290
The Self-Test	291
Warranty	293
Customer Support	294
Regulatory information	298
Federal Communications Commission Notice	298
User Memory and the Stack	301
Managing Calculator Memory	301
Resetting the Calculator	302
Clearing Memory	303
The Status of Stack Lift	304
Disabling Operations	305
Neutral Operations	305
The Status of the LAST X Register	306
Accessing Stack Register Contents	307
ALG: Summary	309
About ALG	309
Doing Two argument Arithmetic in ALG	310
Simple Arithmetic	310
Power Functions	311
Percentage Calculations	311
Permutations and Combinations	312
Quotient and Remainder Of Division	312
Parentheses Calculations	312
Exponential and Logarithmic Functions	313
Trigonometric Functions	314
Hyperbolic functions	314
Parts of numbers	315
Reviewing the Stack	315
Integrating an Equation	316
Operations with Complex Numbers	316
Arithmetic in Bases 2, 8, and 16	318
Entering Statistical Two-Variable Data	319
More about Solving	323
How SOLVE Finds a Root	323
Interpreting Results	325
When SOLVE Cannot Find a Root	330
Round-Off Error	335
More about Integration	337
How the Integral Is Evaluated	337
Conditions That Could Cause Incorrect Results	338
Conditions That Prolong Calculation Time	343
Messages	347
Operation Index	353