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| | Today I am sharing a dark experience with you because I am out of this darkness now. Whoa! Google Chrome has crashed was the disaster that struck on me early this morning. My blood flow seemed to stop for a few minutes. Then I regained my senses and decided to fix Chrome crash.<br><br>Google Chrome crashes on Windows 7 by the corrupted cache contents and problems with all the stored browsing data. Delete the browsing data plus clear the contents of the cache to solve this issue.<br><br>With RegCure to better the start and shut down of your computer. The program shows the scan progress plus you shouldn't worry where it is very working at which time. It shows we what occurs. Dynamic link library section of the registry will result severe application failures. RegCure restores and repairs the registry and keeps you from DLL. RegCure is able to create individual corrections, so it may works for the demands.<br><br>Registry cleaners have been crafted for 1 purpose - to wash out the 'registry'. This is the central database which Windows relies on to function. Without this database, Windows wouldn't even exist. It's so significant, which the computer is continually adding and updating the files inside it, even when you're browsing the Internet (like now). This really is perfect, but the problems happen when certain of those files become corrupt or lost. This happens a lot, plus it takes a advantageous tool to fix it.<br><br>So to fix this, we really should be able to make all of the registry files non-corrupted again. This might dramatically accelerate the loading time of the computer plus allows you to do a big number of items on it again. And fixing these files couldn't be simpler - we merely have to employ a tool called a [http://bestregistrycleanerfix.com/regzooka regzooka].<br><br>Files with the DOC extension are equally susceptible to viruses, nevertheless this could be solved by advantageous antivirus programs. Another problem is that .doc files could be corrupted, unreadable or damaged due to spyware, adware, and malware. These cases usually prevent users from correctly opening DOC files. This really is when effective registry cleaners become valuable.<br><br>Maybe you're asking how come these windows XP error messages appear. Well, for we to be capable to understand the fix, you need to first recognize where those mistakes come from. There is this software called registry. A registry is software which stores everything on the PC from a normal configuration, setting, info, and logs of activities from installing to UN-installing, saving to deleting, along with a lot more alterations we do inside your system pass by it and gets 'tagged' plus saved because a simple file for recovery reasons. Imagine it as a big recorder, a registrar, of all a records in the PC.<br><br>So in summary, when comparing registry cleaning, make certain that that the 1 we choose provides we the following.A backup plus restore facility, fast procedure, automatic deletion center, start-up management, an simple technique of contact along with a money back guarantee. |
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| [[File:Quadratic root.svg|thumb|right|240px|The [[quadratic formula]], which is the solution to the [[quadratic equation]] <math>ax^2+bx+c=0</math> . Here the symbols <math>a,b,c,x</math> all are variables that represent numbers.]]
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| [[File:Polynomialdeg2.svg|thumb|right|200px|Two-dimensional plot (red curve) of the algebraic equation <math>y = x^2 - x - 2</math>]]
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| '''Elementary algebra''' encompasses some of the basic concepts of [[algebra]], one of the main branches of [[mathematics]]. It is typically taught to [[secondary school]] students and builds on their understanding of [[arithmetic]]. Whereas arithmetic deals with specified numbers,<ref>H.E. Slaught and N.J. Lennes, ''Elementary algebra'', Publ. Allyn and Bacon, 1915, [http://books.google.co.uk/books?id=gLii_eO4dNsC&lpg=PA1&dq=%22Elementary%20algebra%22%20letters&pg=PA1#v=onepage&q&f=false page 1] (republished by Forgotten Books)</ref> algebra introduces quantities without fixed values, known as variables.<ref>Lewis Hirsch, Arthur Goodman, ''Understanding Elementary Algebra With Geometry: A Course for College Students'', Publisher: Cengage Learning, 2005, ISBN 0534999727, 9780534999728, 654 pages, [http://books.google.co.uk/books?id=7hdK4RSub5cC&lpg=PA2&dq=what%20is%20algebra%20for&pg=PA2#v=onepage&q=generalization&f=false page 2]</ref> This use of variables entails a use of algebraic notation and an understanding of the general rules of the [[Operation (mathematics)|operator]]s introduced in arithmetic. Unlike [[abstract algebra]], elementary algebra is not concerned with [[algebraic structures]] outside the realm of [[real number|real]] and [[complex number]]s. | |
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| The use of variables to denote quantities allows general relationships between quantities to be formally and concisely expressed, and thus enables solving a broader scope of problems. Most quantitative results in science and mathematics are expressed as algebraic [[equation]]s.
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| ==Algebraic notation ==
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| {{main|Mathematical notation}}
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| Algebraic notation describes how algebra is written. It follows certain rules and conventions, and has its own terminology. For example, the expression <math style="margin-bottom:8px">3x^2 - 2xy + c</math> has the following components:
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| <center>
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| [[File:algebraic equation notation.svg|256px]]<br>
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| 1 : [[Exponent]] (power), 2 : [[Coefficient]], 3 : [[Addend|term]], 4 : [[Operation (mathematics)|operator]], 5 : [[Constant (mathematics)|constant]], <math>x, y</math> : [[Variable (mathematics)|variable]]s
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| </center>
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| A ''coefficient'' is a numerical value which multiplies a variable (the operator is omitted). A ''term'' is an [[Addend#Notation_and_terminology|addend or a summand]], a group of coefficients, variables, constants and exponents that may be separated from the other terms by the plus and minus operators.<ref>Richard N. Aufmann, Joanne Lockwood, ''Introductory Algebra: An Applied Approach'', Publisher Cengage Learning, 2010, ISBN 1439046042, 9781439046043, [http://books.google.co.uk/books?id=MPIWikTHVXQC&lpg=PP1&ots=yG1m9DkIiH&dq=coefficient%20algebra&pg=PA78#v=onepage&q=coefficient%20&f=false page 78]</ref> Letters represent variables and constants. By convention, letters at the beginning of the alphabet (e.g. <math>a, b, c</math>) are typically used to represent [[Mathematical constant|constant]]s, and those toward the end of the alphabet (e.g. <math>x, y</math> and <math>z</math>) are used to represent [[Variable (mathematics)|variable]]s.<ref>William L. Hosch (editor), ''The Britannica Guide to Algebra and Trigonometry'', Britannica Educational Publishing, The Rosen Publishing Group, 2010, ISBN 1615302190, 9781615302192, [http://books.google.co.uk/books?id=ad0P0elU1_0C&lpg=PA71&dq=elementary%20algebra%20letters%20alphabet%20constants%20variables&pg=PA71#v=onepage&q=letters&f=false page 71]</ref> They are usually written in italics.<ref>James E. Gentle, ''Numerical Linear Algebra for Applications in Statistics'', Publisher: Springer, 1998, ISBN 0387985425, 9780387985428, 221 pages, [James E. Gentle page 183]</ref>
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| [[Algebraic operation]]s work in the same way as [[Arithmetic#Algebraic_operations|arithmetic operation]]s,<ref>Horatio Nelson Robinson, ''New elementary algebra: containing the rudiments of science for schools and academies'', Ivison, Phinney, Blakeman, & Co., 1866, [http://books.google.co.uk/books?id=dKZXAAAAYAAJ&dq=Elementary%20algebra%20notation&pg=PA7#v=onepage&q=Elementary%20algebra%20notation&f=false page 7]</ref> such as [[addition]], [[subtraction]], [[multiplication]], [[Division (mathematics)|division]] and [[exponentiation]].<ref>Ron Larson, Robert Hostetler, Bruce H. Edwards, ''Algebra And Trigonometry: A Graphing Approach'', Publisher: Cengage Learning, 2007, ISBN 061885195X, 9780618851959, 1114 pages, [http://books.google.co.uk/books?id=5iXVZHhkjAgC&lpg=PA6&ots=iwrSrCrrOb&dq=operations%20addition%2C%20subtraction%2C%20multiplication%2C%20division%20exponentiation.&pg=PA6#v=onepage&q=operations%20addition,%20subtraction,%20multiplication,%20division%20exponentiation.&f=false page 6]</ref> and are applied to algebraic variables and terms. Multiplication symbols are usually omitted, and implied when there is no space between two variables or terms, or when a [[coefficient]] is used. For example, <math style="margin-bottom:8px">3 \times x^2</math> is written as <math style="margin-bottom:8px">3x^2</math>, and <math>2 \times x \times y</math> may be written <math>2xy</math>.<ref>Sin Kwai Meng, Chip Wai Lung, Ng Song Beng, "Algebraic notation", in ''Mathematics Matters Secondary 1 Express Textbook'', Publisher Panpac Education Pte Ltd, ISBN 9812738827, 9789812738820, [http://books.google.co.uk/books?id=nL5ObMmDvPEC&lpg=PR9-IA8&ots=T_h6l40AE5&dq=%22Algebraic%20notation%22%20multiplication%20omitted&pg=PR9-IA8#v=onepage&q=%22Algebraic%20notation%22%20multiplication%20omitted&f=false page 68]</ref>
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| Usually terms with the highest power ([[Exponentiation|exponent]]), are written on the left, for example, <math style="margin-bottom:8px">x^2</math> is written to the left of <math>x</math>. When a coefficient is one, it is usually omitted (e.g. <math style="margin-bottom:8px">1x^2</math> is written <math style="margin-bottom:8px">x^2</math>).<ref>David Alan Herzog, ''Teach Yourself Visually Algebra'', Publisher John Wiley & Sons, 2008, ISBN 0470185597, 9780470185599, 304 pages, [http://books.google.co.uk/books?id=Igs6t_clf0oC&lpg=PA72&ots=Excnhf1AgW&dq=algebra%20coefficient%20one&pg=PA72#v=onepage&q=coefficient%20of%201&f=false page 72]</ref> Likewise when the exponent (power) is one, (e.g. <math style="margin-bottom:8px">3x^1</math> is written <math style="margin-bottom:8px">3x</math>).<ref>John C. Peterson, ''Technical Mathematics With Calculus'', Publisher Cengage Learning, 2003, ISBN 0766861899, 9780766861893, 1613 pages, [http://books.google.co.uk/books?id=PGuSDjHvircC&lpg=PA31&ots=NKrtZZ1KDE&dq=%22when%20the%20exponent%20is%201%22&pg=PA32#v=onepage&q=%22when%20the%20exponent%20is%201%22&f=false page 31]</ref> When the exponent is zero, the result is always 1 (e.g. <math style="margin-bottom:8px">x^0</math> is always rewritten to <math>1</math>).<ref>Jerome E. Kaufmann, Karen L. Schwitters, ''Algebra for College Students'', Publisher Cengage Learning, 2010, ISBN 0538733543, 9780538733540, 803 pages, [http://books.google.co.uk/books?id=-AHtC0IYMhYC&lpg=PP1&ots=kL8erjajyR&dq=algebra%20exponents%20zero%20one&pg=PA222#v=onepage&q=exponents%20&f=false page 222]</ref> However <math>0^0</math>, being undefined, should not appear in an expression, and care should be taken in simplifying expressions in which variables may appear in exponents.
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| ===Alternative notation===
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| Other types of notation are used in algebraic expressions when the required formatting is not available, or can not be implied, such as where only letters and symbols are available. For example, exponents are usually formatted using superscripts, e.g. <math style="margin-bottom:8px">x^2</math>. In [[plain text]], and in the [[TeX]] mark-up language, the [[caret]] symbol "^" represents exponents, so <math style="margin-bottom:8px">x^2</math> is written as "x^2".<ref>Ramesh Bangia, ''Dictionary of Information Technology'', Publisher Laxmi Publications, Ltd., 2010, ISBN 9380298153, 9789380298153, [http://books.google.co.uk/books?id=zQa5I2sHPKEC&lpg=PA212&ots=s6pWav1Z_D&dq=%22plain%20text%22%20math%20caret%20exponent&pg=PA212#v=onepage&q=exponentiation%20caret&f=false page 212]</ref><ref>George Grätzer, ''First Steps in LaTeX'', Publisher Springer, 1999, ISBN 0817641327, 9780817641320, [http://books.google.co.uk/books?id=mLdg5ZdDKToC&lpg=PP1&ots=V9DFIaAAh0&dq=tex%20math&pg=PA17#v=onepage&q=subscripts%20and%20superscripts%20caret&f=false page 17]</ref> In programming languages such as [[Ada (programming language)|Ada]],<ref>S. Tucker Taft, Robert A. Duff, Randall L. Brukardt, Erhard Ploedereder, Pascal Leroy, ''Ada 2005 Reference Manual'', Volume 4348 of Lecture Notes in Computer Science, Publisher Springer, 2007, ISBN 3540693351, 9783540693352, [http://books.google.co.uk/books?id=694P3YtXh-0C&lpg=PA718&ots=O_EgQ75FeB&dq=ada%20%20asterisk&pg=PA12#v=onepage&q=double%20star%20exponentiate&f=false page 13]</ref> [[Fortran]],<ref>C. Xavier, ''Fortran 77 And Numerical Methods'', Publisher New Age International, 1994, ISBN 812240670X, 9788122406702, [http://books.google.co.uk/books?id=WYMgF9WFty0C&lpg=PA20&ots=BTtzs9F-NB&dq=fortran%20asterisk%20exponentiation&pg=PA20#v=onepage&q=fortran%20asterisk%20exponentiation&f=false page 20]</ref> [[Perl]],<ref>Randal Schwartz, Brian Foy, Tom Phoenix, ''Learning Perl'', Publisher O'Reilly Media, Inc., 2011, ISBN 1449313140, 9781449313142, [http://books.google.co.uk/books?id=l2IwEuRjeNwC&lpg=PA24&ots=5nsYOLHxlD&dq=perl%20asterisk%20exponentiation&pg=PA24#v=onepage&q=double%20asterisk%20exponentiation&f=false page 24]</ref> [[Python (programming language)|Python]] <ref>Matthew A. Telles, ''Python Power!: The Comprehensive Guide'', Publisher Course Technology PTR, 2008, ISBN 1598631586, 9781598631586, [http://books.google.co.uk/books?id=754knV_fyf8C&lpg=PA46&ots=8fEi1F-H8-&dq=python%20asterisk%20exponentiation&pg=PA46#v=onepage&q=double%20asterisk%20exponentiation&f=false page 46]</ref> and [[Ruby (programming language)|Ruby]],<ref>Kevin C. Baird, ''Ruby by Example: Concepts and Code'', Publisher No Starch Press, 2007, ISBN 1593271484, 9781593271480, [http://books.google.co.uk/books?id=kq2dBNdAl3IC&lpg=PA72&ots=0UU3k-Pvh8&dq=ruby%20asterisk%20exponentiation&pg=PA72#v=onepage&q=double%20asterisk%20exponentiation&f=false page 72]</ref> a double asterisk is used, so <math style="margin-bottom:8px">x^2</math> is written as "x**2". Many programming languages and calculators use a single asterisk to represent the multiplication symbol,<ref>William P. Berlinghoff, Fernando Q. Gouvêa, ''Math through the Ages: A Gentle History for Teachers and Others'', Publisher MAA, 2004, ISBN 0883857367, 9780883857366, [http://books.google.co.uk/books?id=JAXNVaPt7uQC&lpg=PA75&ots=-P78Lrz792&dq=calculator%20asterisk%20multiplication&pg=PA75#v=onepage&q=calculator%20asterisk%20multiplication&f=false page 75]</ref> and it must be explicitly used, for example, <math style="margin-bottom:8px">3x</math> is written "3*x".
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| ==Concepts==
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| ===Variables===
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| [[File:Pi-equals-circumference-over-diametre.svg|thumb|right|Example of variables showing the relationship between a circle's diameter and its circumference. For any [[circle]], its [[circumference]] <math>c</math>, divided by its [[diameter]] <math>d</math>, is equal to the constant [[pi]], <math>\pi</math> (approximately 3.14).]]
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| {{Main|Variable (mathematics)}}
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| Elementary algebra builds on and extends arithmetic<ref>Thomas Sonnabend, ''Mathematics for Teachers: An Interactive Approach for Grades K-8'', Publisher: Cengage Learning, 2009, ISBN 0495561665, 9780495561668, 759 pages, [http://books.google.co.uk/books?id=gBa2GzyXCF8C&lpg=PR17&ots=qee3RsTC6V&dq=algebra%20%22extends%20arithmetic%22&pg=PR17#v=onepage&q=extends%20arithmetic&f=false page xvii]</ref> by introducing letters called variables to represent general (non-specified) numbers. This is useful for several reasons.
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| #'''Variables may represent numbers whose values are not yet known'''. For example, if the temperature today, ''T'', is 20 degrees higher than the temperature yesterday, ''Y'', then the problem can be described algebraically as <math>T = Y + 20</math>.<ref>Lewis Hirsch, Arthur Goodman, ''Understanding Elementary Algebra With Geometry: A Course for College Students'', Publisher: Cengage Learning, 2005, ISBN 0534999727, 9780534999728, 654 pages, [http://books.google.co.uk/books?id=jsT7kqZubvIC&lpg=PA48&ots=EI4_yaKasG&dq=%22elementary%20algebra%22%20variables%20unknown&pg=PA48#v=onepage&q=%22elementary%20algebra%22%20variables%20unknown&f=false page 48]</ref>
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| #'''Variables allow one to describe ''general'' problems,<ref>Lawrence S. Leff, ''College Algebra: Barron's Ez-101 Study Keys'', Publisher: Barron's Educational Series, 2005, ISBN 0764129147, 9780764129148, 230 pages, [http://books.google.co.uk/books?id=XesryURrNKAC&lpg=PA2&ots=Ga44CTvNHI&dq=algebra%20variables%20generalize&pg=PA2#v=onepage&q=algebra%20variables%20generalize&f=false page 2]</ref> without specifying the values of the quantities that are involved.''' For example, it can be stated specifically that 5 minutes is equivalent to <math>60 \times 5 = 300</math> seconds. A more general (algebraic) description may state that the number of seconds, <math>s = 60 \times m</math>, where m is the number of minutes.
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| #'''Variables allow one to describe mathematical relationships between quantities that may vary.'''<ref>Ron Larson, Kimberly Nolting, ''Elementary Algebra'', Publisher: Cengage Learning, 2009, ISBN 0547102275, 9780547102276, 622 pages, [http://books.google.co.uk/books?id=U6v78M5nYKAC&lpg=PP1&ots=R0dl97lfm0&dq=%22elementary%20algebra%22%20relationships&pg=PA210#v=onepage&q=relationships&f=false page 210]</ref> For example, the relationship between the circumference, ''c'', and diameter, ''d'', of a circle is described by <math>\pi = c /d</math>.
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| #'''Variables allow one to describe some mathematical properties.''' For example, a basic property of addition is [[commutativity]] which states that the order of numbers being added together does not matter. Commutativity is stated algebraically as <math>(a + b) = (b + a)</math>.<ref>Charles P. McKeague, ''Elementary Algebra'', Publisher: Cengage Learning, 2011, ISBN 0840064217, 9780840064219, 571 pages, [http://books.google.co.uk/books?id=etTbP0rItQ4C&lpg=PA49&ots=I16eebO3LV&dq=%22elementary%20algebra%22%20commutative&pg=PA49#v=onepage&q=%22elementary%20algebra%22%20commutative&f=false page 49]</ref>
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| === Evaluating expressions ===
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| {{Main|Expression (mathematics)}}
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| Algebraic expressions may be evaluated and simplified, based on the basic properties of arithmetic operations ([[addition]], [[subtraction]], [[multiplication]], [[Division (mathematics)|division]] and [[exponentiation]]). For example,
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| *Added terms are simplified using coefficients. For example <math>x + x + x</math> can be simplified as <math>3x</math> (where 3 is the coefficient).
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| *Multiplied terms are simplified using exponents. For example <math>x \times x \times x</math> is represented as <math>x^3</math>
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| *Like terms are added together,<ref>Andrew Marx, ''Shortcut Algebra I: A Quick and Easy Way to Increase Your Algebra I Knowledge and Test Scores'', Publisher Kaplan Publishing, 2007, ISBN 1419552880, 9781419552885, 288 pages, [http://books.google.co.uk/books?id=o9GYQjZ7ZwUC&lpg=PP1&ots=pT-MpWMJty&dq=algebra%20addition%20%22like%20terms%22&pg=PA51#v=onepage&q=like%20terms&f=false page 51]</ref> for example, <math>2x^2 + 3ab - x^2 + ab</math> is written as <math>x^2 + 4ab</math>, because the terms containing <math>x^2</math> are added together, and, the terms containing <math>ab</math> are added together.
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| *Brackets can be "multiplied out", using [[Distributive property|distributivity]]. For example, <math>x (2x + 3)</math> can be written as <math>(x \times 2x) + (x \times 3)</math> which can be written as <math>2x^2 + 3x</math>
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| *Expressions can be factored. For example, <math>6x^5 + 3x^2</math>, by dividing both terms by <math>3x^2</math> can be written as <math>3x^2 (2x^3 + 1)</math>
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| === Equations ===
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| [[File:Pythagorean theorem.gif|thumb|Animation illustrating [[Pythagorean theorem|Pythagoras' rule]] for a right-angle triangle, which shows the algebraic relationship between the triangle's hypotenuse, and the other two sides.]]
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| {{Main|Equation}}
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| An equation states that two expressions are equal using the symbol for equality, <math>=</math> (the [[equals sign]]).<ref>Mark Clark, Cynthia Anfinson, ''Beginning Algebra: Connecting Concepts Through Applications'', Publisher Cengage Learning, 2011, ISBN 0534419380, 9780534419387, 793 pages, [http://books.google.co.uk/books?id=wCzuRMC5048C&lpg=PA134&ots=UdyGuk1ihH&dq=algebra%20equation%20%22two%20expressions%22&pg=PA134#v=onepage&q=equation&f=false page 134]</ref> One of the most well-known equations describes Pythagoras' law relating the length of the sides of a [[right angle]] triangle:<ref>Alan S. Tussy, R. David Gustafson, ''Elementary and Intermediate Algebra'', Publisher Cengage Learning, 2012, ISBN 1111567689, 9781111567682, 1163 pages, [http://books.google.co.uk/books?id=xqio_Xn4t7oC&lpg=PA493&ots=pmzfzBO1KX&dq=algebra%20Pythagoras%20hypotenuse&pg=PA493#v=onepage&q=algebra%20Pythagoras%20hypotenuse&f=false page 493]</ref>
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| :<math>c^2 = a^2 + b^2</math>
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| This equation states that <math>c^2</math>, representing the square of the length of the side that is the hypotenuse (the side opposite the right angle), is equal to the sum (addition) of the squares of the other two sides whose lengths are represented by <math>a</math> and <math>b</math>.
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| An equation is the claim that two expressions have the same value and are equal. Some equations are true for all values of the involved variables (such as <math>a + b = b + a</math>); such equations are called [[identity (mathematics)|identities]]. Conditional equations are true for only some values of the involved variables (e.g. <math>x^2 - 1 = 8</math> is true only for <math>x = 3</math> and <math>x = -3</math>. The values of the variables which make the equation true are the solutions of the equation and can be found through [[equation solving]].
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| Another type of equation is an inequality. Inequalities are used to show that one side of the equation is greater, or less, than the other. The symbols used for this are: <math> a > b </math> where <math> > </math> represents 'greater than', and <math> a < b </math> where <math> < </math> represents 'less than'. Just like standard equality equations, numbers can be added, subtracted, multiplied or divided. The only exception is that when multiplying or dividing by a negative number, the inequality symbol must be flipped.
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| ==== Properties of equality ====
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| By definition, equality is an [[equivalence relation]], meaning it has the properties (a) [[reflexive relation|reflexive]] (i.e. <math>b = b</math>), (b) [[symmetric relation|symmetric]] (i.e. if <math>a = b</math> then <math>b = a</math>) (c) [[transitive relation|transitive]] (i.e. if <math>a = b</math> and <math>b = c</math> then <math>a = c</math>).<ref>Douglas Downing, ''Algebra the Easy Way'', Publisher Barron's Educational Series, 2003, ISBN 0764119729, 9780764119729, 392 pages, [http://books.google.co.uk/books?id=RiX-TJLiQv0C&lpg=PA20&ots=BjArsBq8vc&dq=algebra%20equality%20%20%20reflexive%20%20symmetric%20%20transitive&pg=PA20#v=onepage&q=algebra%20equality%20%20%20reflexive%20%20symmetric%20%20transitive&f=false page 20]</ref> It also satisfies the important property that if two symbols are used for equal things, then one symbol can be substituted for the other in any true statement about the first and the statement will remain true. This implies the following properties:
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| * if <math>a = b</math> and <math>c = d</math> then <math>a + c = b + d</math> and <math>ac = bd</math>;
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| * if <math>a = b</math> then <math>a + c = b + c</math>;
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| * more generally, for any function <math>f</math>, <math>a=b</math> implies <math>f(a) = f(b)</math>.
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| ==== Properties of inequality ====
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| The relations ''less than'' <math> < </math> and greater than <math> > </math> have the property of transitivity:<ref>Ron Larson, Robert Hostetler, ''Intermediate Algebra'', Publisher Cengage Learning, 2008, ISBN 0618753524, 9780618753529, 857 pages, [http://books.google.co.uk/books?id=b3vqad8tbiAC&lpg=PA96&ots=FF2OYYGNCV&dq=algebra%20inequality%20properties&pg=PA96#v=onepage&q=algebra%20inequality%20properties&f=false page 96]</ref>
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| * If <math>a < b</math> and <math>b < c</math> then <math>a < c</math>;
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| * If <math>a < b</math> and <math>c < d</math> then <math>a + c < b + d</math>;
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| * If <math>a < b</math> and <math>c > 0</math> then <math>ac < bc</math>;
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| * If <math>a < b</math> and <math>c < 0</math> then <math>bc < ac</math>.
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| Note that by reversing the equation, we can swap <math> < </math> and <math> > </math>,<ref>Chris Carter, ''Physics: Facts and Practice for A Level'', Publisher Oxford University Press, 2001, ISBN 019914768X, 9780199147687, 144 pages, [http://books.google.co.uk/books?id=Ff9gxZPYafcC&lpg=PA50&ots=KQ5ufGdcHk&dq=algebra%20inequality%20less%20greater%20exchange&pg=PA50#v=onepage&q=turned%20around&f=false page 50]</ref> for example:
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| * <math>a < b</math> is equivalent to <math>b > a</math>
| |
| | |
| == Solving algebraic equations ==
| |
| {{see also|Equation solving}}
| |
| [[File:Algebraproblem.jpg|thumb|A typical algebra problem.]]
| |
| | |
| The following sections lay out examples of some of the types of algebraic equations that may be encountered.
| |
| | |
| === Linear equations with one variable ===
| |
| {{Main|Linear equation}}
| |
| | |
| Linear equations are so-called, because when they are plotted, they describe a straight line (hence ''linear''). The simplest equations to solve are [[linear equation]]s that have only one variable. They contain only constant numbers and a single variable without an exponent. As an example, consider:
| |
| | |
| :Problem in words: If you double my son's age and add 4, the resulting answer is 12. How old is my son?
| |
| | |
| :Equivalent equation: <math>2x + 4 = 12</math> where <math>x</math> represent my son's age
| |
| | |
| To solve this kind of equation, the technique is add, subtract, multiply, or divide both sides of the equation by the same number in order to isolate the variable on one side of the equation. Once the variable is isolated, the other side of the equation is the value of the variable.<ref>{{Cite book|title=All the Math You'll Ever Need|author=Slavin, Steve|publisher=[[John Wiley & Sons]]|year=1989|isbn=0-471-50636-2|page=72}}</ref> This problem and its solution are as follows:
| |
| {|
| |
| |-
| |
| | 1. Equation to solve:
| |
| | <math>2x + 4 = 12</math>
| |
| |-
| |
| | 2. Subtract 4 from both sides:
| |
| | <math>2x + 4 - 4 = 12 - 4</math>
| |
| |-
| |
| | 3. This simplifies to:
| |
| | <math>2x = 8</math>
| |
| |-
| |
| | 4. Divide both sides by 2:
| |
| | <math>\frac{2x}{2} = \frac{8}{2}</math>
| |
| |-
| |
| | 5. Simplifies to the solution:
| |
| | <math>x = 4</math>
| |
| |}
| |
| | |
| The general form of a linear equation with one variable, can be written as: <math>ax+b=c\,</math>
| |
| | |
| Following the same procedure (i.e. subtract <math>b</math> from both sides, and then divide by <math>a</math>), the general solution is given by <math>x=\frac{c-b}{a}</math>
| |
| | |
| === Linear equations with two variables ===
| |
| [[File:Linear-equations-two-unknowns.svg|thumb|right|Two linear equations|Solving two linear equations with a unique solution at the point that they intersect.]]
| |
| A linear equation with two variables has many (i.e. an infinite number of) solutions.<ref>Sinha, ''The Pearson Guide to Quantitative Aptitude for CAT 2/e''Publisher: Pearson Education India, 2010, ISBN 8131723666, 9788131723661, 599 pages, [http://books.google.co.uk/books?id=eOnaFSKRSR0C&lpg=PA195&ots=K9A5d10dUc&dq=linear%20equation%20%20two%20variables%20%20many%20solutions&pg=PA195#v=onepage&q=many%20solutions&f=false page 195]</ref> For example:
| |
| | |
| :Problem in words: I am 22 years older than my son. How old are we?
| |
| :Equivalent equation: <math>y = x + 22</math> where <math>y</math> is my age, <math>x</math> is my son's age.
| |
| | |
| This can not be worked out by itself. If I told you my son's age, then there would no longer be two unknowns (variables), and the problem becomes a linear equation with just one variable, that can be solved as described above.
| |
| | |
| To solve a linear equation with two variables (unknowns), requires two related equations. For example, if I also revealed that:
| |
| {|
| |
| |-
| |
| | Problem in words:
| |
| | In 10 years time, I will be twice as old as my son.
| |
| |-
| |
| | Equivalent equation:
| |
| | <math>y + 10 = 2 \times (x + 10)</math>
| |
| |-
| |
| | Subtract 10 from both sides:
| |
| | <math>y = 2 \times (x + 10) - 10</math>
| |
| |-
| |
| | Multiple out brackets:
| |
| | <math>y = 2x + 20 - 10</math>
| |
| |-
| |
| | Simplify:
| |
| | <math>y = 2x + 10</math>
| |
| |}
| |
| | |
| Now there are two related linear equations, each with two unknowns, which lets us produce a linear equation with just one variable, by subtracting one from the other (called the elimination method):<ref>Cynthia Y. Young, ''Precalculus'', Publisher John Wiley & Sons, 2010, ISBN 0471756849, 9780471756842, 1175 pages, [http://books.google.co.uk/books?id=9HRLAn326zEC&lpg=PA703&ots=t83cFBL8TU&dq=linear%20equation%20%20two%20variables%20%20many%20solutions&pg=PA699#v=onepage&q=linear%20equation%20%20two%20variables%20%20many%20solutions&f=false page 699]</ref>
| |
| {|
| |
| |-
| |
| | Second equation
| |
| | <math>y = 2x + 10</math>
| |
| |-
| |
| | First equation
| |
| | <math>y = x + 22</math>
| |
| |-
| |
| | Subtract the first equation from<br>the second in order to remove <math>y</math>
| |
| | <math>(y - y) = (2x - x) +10 - 22</math>
| |
| |-
| |
| | Simplify
| |
| | <math>0 = x - 12</math>
| |
| |-
| |
| | Add 12 to both sides
| |
| | <math>12 = x</math>
| |
| |-
| |
| | Rearrange
| |
| | <math>x = 12</math>
| |
| |}
| |
| | |
| In other words, my son is aged 12, and as I am 22 years older, I must be 34. In 10 years time, my son will 22, and I will be twice his age, 44. This problem is illustrated on the associated plot of the equations.
| |
| | |
| For other ways to solve this kind of equations, see below, '''[[#System of linear equations|System of linear equations]]'''.
| |
| | |
| === Quadratic equations ===
| |
| {{Main|Quadratic equation}}
| |
| [[File:Quadratic-equation.svg|thumb|right|Quadratic equation plot of <math>y = x^2 + 3x - 10</math> showing its roots at <math>x = -5</math> and <math>x = 2</math>, and that the quadratic can be rewritten as <math>y = (x + 5)(x - 2)</math>
| |
| ]]
| |
| A quadratic equation is one which includes a term with an exponent of 2, for example, <math>x^2</math>,<ref>Mary Jane Sterling, ''Algebra II For Dummies'', Publisher: John Wiley & Sons, 2006, ISBN 0471775819, 9780471775812, 384 pages, [http://books.google.co.uk/books?id=_0rTMuSpTY0C&lpg=PA43&ots=J8_1q1Vkul&dq=quadratic%20equations&pg=PA37#v=onepage&q=quadratic%20equations&f=false page 37]</ref> and no term with higher exponent. The name derives from the Latin ''quadrus'', meaning square.<ref>John T. Irwin, ''The Mystery to a Solution: Poe, Borges, and the Analytic Detective Story'', Publisher JHU Press, 1996, ISBN 0801854660, 9780801854668, 512 pages, [http://books.google.co.uk/books?id=jsxTenuOQKgC&lpg=PA372&ots=T7p7Porq55&dq=quadratic%20quadrus&pg=PA372#v=onepage&q=quadratic%20quadrus&f=false page 372]</ref> In general, a quadratic equation can be expressed in the form <math>ax^2 + bx + c = 0</math>,<ref>Sharma/khattar, ''The Pearson Guide To Objective Mathematics For Engineering Entrance Examinations, 3/E'', Publisher Pearson Education India, 2010, ISBN 8131723631, 9788131723630, 1248 pages, [http://books.google.co.uk/books?id=2v-f9x7-FlsC&lpg=RA13-PA33&ots=P-Dz70fXbv&dq=quadratic%20equations%20%20ax2%20%2B%20bx%20%2B%20c%20%3D%200&pg=RA13-PA33#v=onepage&q&f=false page 621]</ref> where <math>a</math> is not zero (if it were zero, then the equation would not be quadratic but linear). Because of this a quadratic equation must contain the term <math>ax^2</math>, which is known as the quadratic term. Hence <math>a \neq 0</math>, and so we may divide by <math>a</math> and rearrange the equation into the standard form
| |
| | |
| : <math>x^2 + px + q = 0 \, </math>
| |
| | |
| where <math>p = b/a</math> and <math>q = c/a</math>. Solving this, by a process known as [[completing the square]], leads to the [[quadratic formula]]
| |
| | |
| :<math>x=\frac{-b \pm \sqrt {b^2-4ac}}{2a},</math>
| |
| | |
| where [[plus-minus sign|the symbol "±"]] indicates that both
| |
| | |
| : <math> x=\frac{-b + \sqrt {b^2-4ac}}{2a}\quad\text{and}\quad x=\frac{-b - \sqrt {b^2-4ac}}{2a}</math>
| |
| | |
| are solutions of the quadratic equation.
| |
| | |
| Quadratic equations can also be solved using [[factorization]] (the reverse process of which is [[polynomial expansion|expansion]], but for two [[linear function|linear terms]] is sometimes denoted [[FOIL rule|foiling]]). As an example of factoring:
| |
| | |
| : <math>x^{2} + 3x - 10 = 0. \,</math>
| |
| | |
| Which is the same thing as
| |
| | |
| : <math>(x + 5)(x - 2) = 0. \,</math>
| |
| | |
| It follows from the [[zero-product property]] that either <math>x = 2</math> or <math>x = -5</math> are the solutions, since precisely one of the factors must be equal to [[zero]]. All quadratic equations will have two solutions in the [[complex number]] system, but need not have any in the [[real number]] system. For example,
| |
| | |
| : <math>x^{2} + 1 = 0 \,</math>
| |
| | |
| has no real number solution since no real number squared equals −1.
| |
| Sometimes a quadratic equation has a root of [[multiplicity (mathematics)|multiplicity]] 2, such as:
| |
| | |
| : <math>(x + 1)^2 = 0. \,</math>
| |
| | |
| For this equation, −1 is a root of multiplicity 2. This means −1 appears two times.
| |
| | |
| === Exponential and logarithmic equations ===
| |
| {{Main|Logarithm}}
| |
| [[File:Binary logarithm plot with ticks.svg|right|thumb|upright=1.35|alt=Graph showing a logarithm curves, which crosses the ''x''-axis where ''x'' is 1 and extend towards minus infinity along the ''y''-axis.|The [[graph of a function|graph]] of the logarithm to base 2 crosses the [[x axis|''x'' axis]] (horizontal axis) at 1 and passes through the points with [[coordinate]]s {{nowrap|(2, 1)}}, {{nowrap|(4, 2)}}, and {{nowrap|(8, 3)}}. For example, {{nowrap|log<sub>2</sub>(8) {{=}} 3}}, because {{nowrap|2<sup>3</sup> {{=}} 8.}} The graph gets arbitrarily close to the ''y'' axis, but [[asymptotic|does not meet or intersect it]].]]
| |
| | |
| An exponential equation is one which has the form <math>a^x = b</math> for <math>a > 0</math>,<ref>Aven Choo, ''LMAN OL Additional Maths Revision Guide 3'', Publisher Pearson Education South Asia, 2007, ISBN 9810600011, 9789810600013, [http://books.google.co.uk/books?id=NsBXDMrzcJIC&lpg=RA2-PA29&ots=WmZmHLaTey&dq=%22%20exponential%20equation%20%22%20aX%20%3D%20b&pg=RA2-PA29#v=onepage&q=%22%20exponential%20equation%20%22%20aX%20=%20b&f=false page 105]</ref> which has solution
| |
| | |
| : <math>X = \log_a b = \frac{\ln b}{\ln a}</math>
| |
| | |
| when <math>b > 0</math>. Elementary algebraic techniques are used to rewrite a given equation in the above way before arriving at the solution. For example, if
| |
| | |
| : <math>3 \cdot 2^{x - 1} + 1 = 10</math>
| |
| | |
| then, by subtracting 1 from both sides of the equation, and then dividing both sides by 3 we obtain
| |
| | |
| : <math>2^{x - 1} = 3\,</math>
| |
| | |
| whence
| |
| | |
| : <math>x - 1 = \log_2 3\,</math>
| |
| | |
| or
| |
| | |
| : <math>x = \log_2 3 + 1.\,</math>
| |
| | |
| A logarithmic equation is an equation of the form <math>log_a(x) = b</math> for <math>a > 0</math>, which has solution
| |
| | |
| : <math>X = a^b.\,</math>
| |
| | |
| For example, if
| |
| | |
| : <math>4\log_5(x - 3) - 2 = 6\,</math>
| |
| | |
| then, by adding 2 to both sides of the equation, followed by dividing both sides by 4, we get
| |
| | |
| : <math>\log_5(x - 3) = 2\,</math>
| |
| | |
| whence
| |
| | |
| : <math>x - 3 = 5^2 = 25\,</math>
| |
| | |
| from which we obtain
| |
| | |
| : <math>x = 28.\,</math>
| |
| | |
| === Radical equations ===
| |
| [[File:Radical equation equivalence.svg|thumb|right|Radical equation showing two ways to represent the same expression]]
| |
| A radical equation is one that includes a radical sign, which includes [[square root]]s, <math>\sqrt{x}</math>, [[cube root]]s, <math>\sqrt[3]{x}</math>, and [[nth root|''n''th roots]], <math>\sqrt[n]{x}</math>. Recall that an ''n''th root can be rewritten in exponential format, so that <math>\sqrt[n]{x}</math> is equivalent to <math>x^{\frac{1}{n}}</math>. Combined with regular exponents (powers), then <math>\sqrt[2]{x^3}</math> (the square root of <math>x</math> cubed), can be rewritten as <math>x^{\frac{3}{2}}</math>.<ref>John C. Peterson, ''Technical Mathematics With Calculus'', Publisher Cengage Learning, 2003, ISBN 0766861899, 9780766861893, 1613 pages, [http://books.google.co.uk/books?id=PGuSDjHvircC&lpg=PA525&ots=NKrt__3KKE&dq=%22%20radical%20equation%22&pg=PA525#v=onepage&q=%22%20radical%20equation%22&f=false page 525]</ref> So the general form of radical equation is <math>a = \sqrt[n]{x^m}</math> (equivalent to <math>a = x^\frac{m}{n}</math>) where <math>m</math> and <math>n</math> [[integer]]s, and which has solution:
| |
| {| class="wikitable" style="text-align:center"
| |
| |- style="vertical-align:top"
| |
| ! <math>m</math> is odd
| |
| ! <math>m</math> is even<br>and <math>a \ge 0</math>
| |
| |-
| |
| | <math>x = \sqrt[m]{a^n}</math>
| |
| | |
| or
| |
| | |
| : <math>x = \left(\sqrt[m]a\right)^n</math>
| |
| | <math>x = \pm \sqrt[m]{a^n}</math>
| |
| | |
| or
| |
| | |
| : <math>x = \pm \left(\sqrt[m]a\right)^n</math>
| |
| |}
| |
| | |
| For example, if:
| |
| | |
| : <math>(x + 5)^{2/3} = 4,\,</math>
| |
| | |
| then
| |
| | |
| : <math>\begin{align}
| |
| x + 5 & = \pm (\sqrt{4})^3\\
| |
| x + 5 & = \pm 8\\
| |
| x & = -5 \pm 8\\
| |
| x & = 3,-13
| |
| \end{align}</math>.
| |
| | |
| === System of linear equations ===
| |
| {{Main|System of linear equations}}
| |
| | |
| There are different methods to solve a system of linear equations with two variables.
| |
| | |
| ==== Elimination method ====
| |
| [[File:Intersecting Lines.svg|thumb|right|The solution set for the equations <math>x - y = -1</math> and <math>3x + y = 9</math> is the single point (2, 3).]]
| |
| | |
| An example of solving a system of linear equations is by using the elimination method:
| |
| | |
| : <math>\begin{cases}4x + 2y&= 14 \\
| |
| 2x - y&= 1.\end{cases} \,</math>
| |
| | |
| Multiplying the terms in the second equation by 2:
| |
| | |
| : <math>4x + 2y = 14 \,</math>
| |
| : <math>4x - 2y = 2. \,</math>
| |
| | |
| Adding the two equations together to get:
| |
| | |
| : <math>8x = 16 \,</math>
| |
| | |
| which simplifies to
| |
| | |
| : <math>x = 2. \,</math>
| |
| | |
| Since the fact that <math>x = 2</math> is known, it is then possible to deduce that <math>y = 3</math> by either of the original two equations (by using ''2'' instead of <math>x</math> ) The full solution to this problem is then
| |
| | |
| : <math>\begin{cases} x = 2 \\ y = 3. \end{cases}\,</math>
| |
| | |
| Note that this is not the only way to solve this specific system; <math>y</math> could have been solved before <math>x</math>.
| |
| | |
| ==== Substitution method ====
| |
| | |
| Another way of solving the same system of linear equations is by substitution.
| |
| | |
| : <math>\begin{cases}4x + 2y &= 14
| |
| \\ 2x - y &= 1.\end{cases} \,</math>
| |
| | |
| An equivalent for <math>y</math> can be deduced by using one of the two equations. Using the second equation:
| |
| | |
| : <math>2x - y = 1 \,</math>
| |
| | |
| Subtracting <math>2x</math> from each side of the equation:
| |
| | |
| : <math>\begin{align}2x - 2x - y & = 1 - 2x \\
| |
| - y & = 1 - 2x
| |
| \end{align}</math>
| |
| | |
| and multiplying by −1:
| |
| | |
| : <math> y = 2x - 1. \,</math>
| |
| | |
| Using this <math>y</math> value in the first equation in the original system:
| |
| | |
| : <math>\begin{align}4x + 2(2x - 1) &= 14\\
| |
| 4x + 4x - 2 &= 14 \\
| |
| 8x - 2 &= 14 \end{align}</math>
| |
| | |
| Adding ''2'' on each side of the equation:
| |
| | |
| : <math>\begin{align}8x - 2 + 2 &= 14 + 2 \\
| |
| 8x &= 16 \end{align}</math>
| |
| | |
| which simplifies to
| |
| | |
| : <math>x = 2 \,</math>
| |
| | |
| Using this value in one of the equations, the same solution as in the previous method is obtained.
| |
| | |
| : <math>\begin{cases} x = 2 \\ y = 3. \end{cases}\,</math>
| |
| | |
| Note that this is not the only way to solve this specific system; in this case as well, <math>y</math> could have been solved before <math>x</math>.
| |
| | |
| === Other types of systems of linear equations ===
| |
| | |
| ==== Unsolvable systems ====
| |
| [[File:Parallel Lines.svg|thumb|right|The equations <math>3x + 2y = 6</math> and <math>3x + 2y = 12</math> are parallel and cannot intersect, and is unsolvable.]]
| |
| In the above example, it is possible to find a solution. However, there are also systems of equations which do not have a solution. An obvious example would be:
| |
| | |
| : <math>\begin{cases}\begin{align} x + y &= 1 \\
| |
| 0x + 0y &= 2 \end{align} \end{cases}\,</math>
| |
| | |
| The second equation in the system has no possible solution. Therefore, this system can't be solved.
| |
| However, not all incompatible systems are recognized at first sight. As an example, the following system is studied:
| |
| | |
| : <math>\begin{cases}\begin{align}4x + 2y &= 12 \\
| |
| -2x - y &= -4 \end{align}\end{cases}\,</math>
| |
| | |
| When trying to solve this (for example, by using the method of substitution above), the second equation, after adding <math>2x</math> on both sides and multiplying by −1, results in:
| |
| | |
| : <math>y = -2x + 4 \,</math>
| |
| | |
| And using this value for <math>y</math> in the first equation:
| |
| | |
| : <math>\begin{align}4x + 2(-2x + 4) &= 12 \\
| |
| 4x - 4x + 8 &= 12 \\
| |
| 8 &= 12 \end{align}</math>
| |
| | |
| No variables are left, and the equality is not true. This means that the first equation can't provide a solution for the value for <math>y</math> obtained in the second equation.
| |
| | |
| ==== Undetermined systems ====
| |
| [[File:Quadratic-linear-equations.svg|thumb|right|Plot of a quadratic equation (red) and a linear equation (blue) that do not intersect, and consequently for which there is no common solution.]]
| |
| There are also systems which have infinitely many solutions, in contrast to a system with a unique solution (meaning, a unique pair of values for <math>x</math> and <math>y</math>) For example:
| |
| | |
| : <math>\begin{cases}\begin{align}4x + 2y & = 12 \\
| |
| -2x - y & = -6 \end{align}\end{cases}\,</math>
| |
| | |
| Isolating <math>y</math> in the second equation:
| |
| | |
| : <math>y = -2x + 6 \,</math>
| |
| | |
| And using this value in the first equation in the system:
| |
| | |
| : <math>\begin{align}4x + 2(-2x + 6) = 12 \\
| |
| 4x - 4x + 12 = 12 \\
| |
| 12 = 12 \end{align}</math>
| |
| | |
| The equality is true, but it does not provide a value for <math>x</math>. Indeed, one can easily verify (by just filling in some values of <math>x</math>) that for any <math>x</math> there is a solution as long as <math>y = -2x + 6</math>. There is an infinite number of solutions for this system.
| |
| | |
| ==== Over- and underdetermined systems ====
| |
| | |
| Systems with more variables than the number of linear equations do not have a unique solution. An example of such a system is
| |
| | |
| : <math>\begin{cases}\begin{align}x + 2y & = 10\\
| |
| y - z & = 2 \end{align}\end{cases}</math>
| |
| | |
| Such a system is called [[Underdetermined system|underdetermined]]; when trying to solve it, one is lead to express some variables as functions of the other ones, but cannot express ''all'' solutions [[Number|numerically]].
| |
| Incidentally, a system with a greater number of equations than variables, in which necessarily some equations are [[linear combination]] of the others, is called [[Overdetermined system|overdetermined]].
| |
| | |
| == See also ==
| |
| * [[History of elementary algebra]]
| |
| * [[Binary operation]]
| |
| * [[Gaussian elimination]]
| |
| * [[Mathematics education]]
| |
| * [[Number line]]
| |
| * [[Polynomial]]
| |
| | |
| == References ==
| |
| | |
| * [[Leonhard Euler]], '' [[Elements of Algebra]]'', 1770. English translation [[Tarquin Press]], 2007, ISBN 978-1-899618-79-8, also online digitized editions<ref>[http://web.mat.bham.ac.uk/C.J.Sangwin/euler/ Euler's Elements of Algebra<!-- Bot generated title -->]</ref> 2006,<ref>[http://books.google.co.uk/books?id=X8yv0sj4_1YC&dq=euler+elements&psp=1 Elements of algebra – Leonhard Euler, John Hewlett, Francis Horner, Jean Bernoulli, Joseph Louis Lagrange – Google Books<!-- Bot generated title -->]</ref> 1822.
| |
| * Charles Smith, ''[http://digital.library.cornell.edu/cgi/t/text/text-idx?c=math;idno=smit025 A Treatise on Algebra]'', in [http://historical.library.cornell.edu/math Cornell University Library Historical Math Monographs].
| |
| *Redden, John. [http://catalog.flatworldknowledge.com/bookhub/reader/128?e=fwk-redden-ch01 ''Elementary Algebra'']. Flat World Knowledge, 2011
| |
| | |
| {{reflist}}
| |
| | |
| == External links ==
| |
| | |
| {{Mathematics-footer}}
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| {{DEFAULTSORT:Elementary Algebra}}
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| [[Category:Algebra]]
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| [[Category:Elementary algebra|*]]
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