Maxima by Example: Ch.4: Solving Equations ? PDF

Solving equations: linear quadratic

simultaneous linear

Solving linear and quadratic simultaneous equations

Use the linear equation to substitute into the quadratic equation. • There are usually two pairs of solutions. Examples. Example 1 Solve the simultaneous

09_EQUATION/FUNCTION_Quick Start Guide (fx-991EX/fx-570EX)

The fx-991EX has the power to handle Simultaneous Equations with up to 4 unknowns and Polynomial Equations up to the 4th degree. SIMULTANEOUS EQUATIONS.

Chapter 2.pmd

Review the concepts of coordinate geometry done earlier including graphs of linear equations. Awareness of geometrical representation of quadratic polynomials.

Algebra

manipulating them to form valid mathematical expressions simultaneous equations

Maxima by Example: Ch.4: Solving Equations ?

29 janv. 2009 4.1.3 General Quadratic Equation or Function . ... linsolve solves a system of simultaneous linear equations for the speci ed variables and ...

The Major Topics of School Algebra

12 juin 2008 Graphing and solving systems of simultaneous linear equations. Quadratic Equations. • Factors and factoring of quadratic polynomials with ...

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4G Solving linear simultaneous equations algebraically . 5a Solving quadratic equations . ... dividing a quadratic polynomial by a linear expression.

Solving Systems of Polynomial Equations

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In this Workbook you will learn about solving single equations mainly linear and quadratic but also cubic and higher degree and also simultaneous linear

[PDF] Solving linear and quadratic simultaneous equations

Use the linear equation to substitute into the quadratic equation • There are usually two pairs of solutions Examples Example 1 Solve the simultaneous

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A polynomial is a function P with a general form The linear equation ax + b = 0 has solution x = b/a This is known as a quadratic polynomial The

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Solve each linear and quadratic system BY SUBSTITUTION State the solution(s) on the line Must SHOW WORK! 5 ) ? ? ?

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We will now discuss the technique of finding the solutions of a system of simultaneous linear inequations By the term solution of a system of simultaneons

[PDF] Solving Linear and Quadratic Equations - The University of Sydney

The important thing to note about quadratic functions is the presence of a squared variable Hence solving a simultaneous equation will require

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This is easily checked by substitution These values are called the solutions of the equation Linear equations that are written in the standard form ax + b =

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How to solve simultaneous equations with linear and quadratic equations?
When solving simultaneous equations with a linear and quadratic equation, there will usually be two pairs of answers. Substitute y = x + 3 into the quadratic equation to create an equation which can be factorised and solved. If the product of two brackets is zero, then one or both brackets must also be equal to zero.
What is the difference between linear equation and quadratic equation and polynomial?
A linear equation produces a straight line when we graph it whereas when we graph a quadratic equation we produce a parabola. The slope of a quadratic polynomial unlike the slope of a linear polynomial, is constantly changing.
How do you solve simultaneous polynomial equations?
To solve a set of simultaneous equations you need to:
1Eliminate one of the variables.2Find the value of one variable.3Find the value of the remaining variables via substitution.4Clearly state the final answer/s.5Check your answer by substituting both values into either of the original equations.
Linear, quadratic and cubic polynomials can be classified on the basis of their degrees.
1A polynomial of degree one is a linear polynomial. For example, 5x + 3.2A polynomial of degree two is a quadratic polynomial. For example, 2x2 + x + 5.3A polynomial of degree three is a cubic polynomial.

Maxima by Example: Ch.4: Solving Equations ?

4.1.1 The Maxima Function solve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

4.1.2 solve with Expressions or Functions & the multiplicities List . . . . . . . . . . . . . . . . . . . . 4

4.1.3 General Quadratic Equation or Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

4.1.4 Checking Solutions with subst or ev and a "Do Loop" . . . . . . . . . . . . . . . . . . . . . . . 6

4.1.5 The One Argument Form of solve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4.1.6 Using disp, display, and print . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4.1.7 Checking Solutions using map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4.1.8 Psuedo-PostFix Code: %% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4.1.9 Using an Expression Rather than a Function with Solve . . . . . . . . . . . . . . . . . . . . . . . 9

4.1.10 Escape Speed from the Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.1.11 Cubic Equation or Expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.1.12 Trigonometric Equation or Expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.1.13 Equation or Expression Containing Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . 15

4.2 One Equation Numerical Solutions: allroots, realroots, nd

root . . . . . . . . . . . . . . . . . . . . . . 16

4.2.1 Comparison of realroots with allroots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.2.2 Intersection Points of Two Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.2.3 Transcendental Equations and Roots: nd

root . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.2.4 nd

root: Quote that Function! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.2.5 newton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.3 Two or More Equations: Symbolic and Numerical Solutions . . . . . . . . . . . . . . . . . . . . . . . . 28

4.3.1 Numerical or Symbolic Linear Equations with solve or linsolve . . . . . . . . . . . . . . . . . . 28

4.3.2 Matrix Methods for Linear Equation Sets: linsolve

lu . . . . . . . . . . . . . . . . . . . . . . 29

4.3.3 Symbolic Linear Equation Solutions: Matrix Methods . . . . . . . . . . . . . . . . . . . . . . . 30

4.3.4 Multiple Solutions from Multiple Right Hand Sides . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.3.5 Three Linear Equation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.3.6 Surpressing rat Messages: ratprint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.3.7 Non-Linear Polynomial Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.3.8 General Sets of Nonlinear Equations: eliminate, mnewton . . . . . . . . . . . . . . . . . . . . . 37

4.3.9 Intersections of Two Circles: implicit

plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.3.10 Using Draw for Implicit Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.3.11 Another Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.3.12 Error Messages and Do It Yourself Mnewton . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.3.13 Automated Code for mymnewton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

This document is part of a series of notes titled "Maxima by Example" and is made available NON-PROFIT PRINTING AND DISTRIBUTION IS PERMITTED.

You may make copies of this document and distribute them to others as long as you charge no more than the

costs of printing. These notes (with some modications) will be published in book form eventually via Lulu.com in an ar- a low cost paperbound version of these notes. This chapter gives examples of the following Maxima functions: We also encourage the use of two dimensional plots to approximately locate solutions.

This chapter does not yet include "Solving Recurrence Relations", and "Solving One Hundred Equations".

Let's use conservation of mechanical energy (kinetic plus potential) to rst calculate the initial radial speed

a rocket must have near the surface of the earth to achieve a nal required radial speed far from the earth (far

We can make a simple plot of our expression to see the periodic behavior and the approximate location of

the real roots.-4 -6-4-2 0 2 4 6 x0.0 sin(x)2-2*sin(x)-3

Figure 1: plot of ex3

(%i1) 2 *(a*log(x) + 2*a*log(y))$ (%i2) logcontract(%); 2 4 (%o2) a log(x y ) root x2¡x¡1 -10

0 1 2 3 4 5

xhx rx=hx - kx

Figure 2: Intersection Points are Zeroes of rx

Here is code you can use to make something close to the above plot.

To nd the roots of transcendental expressions, for example, we can rst make a plot of the expression, and

then use??? root??? ?? ?? root error root abs root rel -0.8 -0.6 -0.4 -0.2 0.2 0.4 0.6

0 0.2 0.4 0.6 0.8 1

plot of x - cos(x)

Figure 3: Plot of x - cos(x)

We can use either an expression or function as the entry to the rst slot of??? root? ? ??? ???? ??? ???? f(x) = cos(x=¼)e¡(x=4)2¡sin(x3=2)¡5=4 ???? ?? ??? ???? ??f(x)?-2.5 -1.5 -0.5 0.5

0 1 2 3 4 5

plot of f(x)

Figure 4: Plot of f(x)

root: Quote that Function! ???root?? ??? root?? ???

The important thing to stress is that the???

2y dy?

root?

The following methods succeed.

i+1=xi¡f(xi)

0(xi):

ax+by=c;dx+ey=f A simple numerical (rather than symbolic) two equation example: lu? If we re-cast the two equation problem we have just been solving in the form of a matrix equation

One should always check solutions when using computer algebra software, since the are occasional bugs in

ax+by=c; dx+ey=f; y=¡x=3: plot plot??? -3-2-1 0 1 2 3 4 5 6

Intersections of Two Circles

Figure 5: two circles

-0.5 0.5 1.5 -3-2-1 0 1 2 3 dennis and schnabel example

Figure 6: Dennis and Schnabel Example

root. We rewrite the equations as expressions here.quotesdbs_dbs7.pdfusesText_5

[PDF] Maxima by Example: Ch.4: Solving Equations ?

How to solve simultaneous equations with linear and quadratic equations?

What is the difference between linear equation and quadratic equation and polynomial?

How do you solve simultaneous polynomial equations?

To solve a set of simultaneous equations you need to:

Linear, quadratic and cubic polynomials can be classified on the basis of their degrees.

4.1.1 The Maxima Function solve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

4.1.2 solve with Expressions or Functions & the multiplicities List . . . . . . . . . . . . . . . . . . . . 4

4.1.3 General Quadratic Equation or Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

4.1.4 Checking Solutions with subst or ev and a "Do Loop" . . . . . . . . . . . . . . . . . . . . . . . 6

4.1.5 The One Argument Form of solve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4.1.6 Using disp, display, and print . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4.1.7 Checking Solutions using map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4.1.8 Psuedo-PostFix Code: %% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4.1.9 Using an Expression Rather than a Function with Solve . . . . . . . . . . . . . . . . . . . . . . . 9

4.1.10 Escape Speed from the Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.1.11 Cubic Equation or Expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.1.12 Trigonometric Equation or Expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.1.13 Equation or Expression Containing Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . 15

4.2 One Equation Numerical Solutions: allroots, realroots, nd

4.2.1 Comparison of realroots with allroots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.2.2 Intersection Points of Two Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.2.3 Transcendental Equations and Roots: nd

4.2.4 nd

4.2.5 newton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.3 Two or More Equations: Symbolic and Numerical Solutions . . . . . . . . . . . . . . . . . . . . . . . . 28

4.3.1 Numerical or Symbolic Linear Equations with solve or linsolve . . . . . . . . . . . . . . . . . . 28

4.3.2 Matrix Methods for Linear Equation Sets: linsolve

4.3.3 Symbolic Linear Equation Solutions: Matrix Methods . . . . . . . . . . . . . . . . . . . . . . . 30

4.3.4 Multiple Solutions from Multiple Right Hand Sides . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.3.5 Three Linear Equation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.3.6 Surpressing rat Messages: ratprint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.3.7 Non-Linear Polynomial Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.3.8 General Sets of Nonlinear Equations: eliminate, mnewton . . . . . . . . . . . . . . . . . . . . . 37

4.3.9 Intersections of Two Circles: implicit

4.3.10 Using Draw for Implicit Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.3.11 Another Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.3.12 Error Messages and Do It Yourself Mnewton . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.3.13 Automated Code for mymnewton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Figure 1: plot of ex3

0 1 2 3 4 5

Figure 2: Intersection Points are Zeroes of rx

0 0.2 0.4 0.6 0.8 1

Figure 3: Plot of x - cos(x)

0 1 2 3 4 5

Figure 4: Plot of f(x)

The important thing to stress is that the???

2y dy?

The following methods succeed.

0(xi):

Intersections of Two Circles

Figure 5: two circles

Figure 6: Dennis and Schnabel Example