Elementary algebra#Second method of finding a solution

{{excerpt|Algebraic operation}}

{{main|Mathematical notation}}

Algebraic notation describes the rules and conventions for writing mathematical expressions, as well as the terminology used for talking about parts of expressions. For example, the expression $3x^2\; -\; 2xy\; +\; c$ has the following components:

File:algebraic equation notation.svg

A coefficient is a numerical value, or letter representing a numerical constant, that multiplies a variable (the operator is omitted). A term is an 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.Richard N. Aufmann, Joanne Lockwood, Introductory Algebra: An Applied Approach, Publisher Cengage Learning, 2010, {{ISBN|1439046042}}, 9781439046043, [https://books.google.com/books?id=MPIWikTHVXQC&q=coefficient+&pg=PA78 page 78] Letters represent variables and constants. By convention, letters at the beginning of the alphabet (e.g. $a,\; b,\; c$) are typically used to represent constants, and those toward the end of the alphabet (e.g. $x,\; y$ and {{mvar|z}}) are used to represent variables.William L. Hosch (editor), The Britannica Guide to Algebra and Trigonometry, Britannica Educational Publishing, The Rosen Publishing Group, 2010, {{ISBN|1615302190}}, 9781615302192, [https://books.google.com/books?id=ad0P0elU1_0C&q=letters&pg=PA71 page 71] They are usually printed in italics.James E. Gentle, Numerical Linear Algebra for Applications in Statistics, Publisher: Springer, 1998, {{ISBN|0387985425}}, 9780387985428, 221 pages, [James E. Gentle page 184]

Algebraic operations work in the same way as arithmetic operations,Horatio Nelson Robinson, New elementary algebra: containing the rudiments of science for schools and academies, Ivison, Phinney, Blakeman, & Co., 1866, [https://books.google.com/books?id=dKZXAAAAYAAJ&dq=Elementary+algebra+notation&pg=PA7 page 7] such as addition, subtraction, multiplication, division and exponentiation,Ron Larson, Robert Hostetler, Bruce H. Edwards, Algebra And Trigonometry: A Graphing Approach, Publisher: Cengage Learning, 2007, {{ISBN|061885195X}}, 9780618851959, 1114 pages, [https://books.google.com/books?id=5iXVZHhkjAgC&dq=operations+addition%2C+subtraction%2C+multiplication%2C+division+exponentiation.&pg=PA6 page 6] 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, $3\; \backslash times\; x^2$ is written as $3x^2$, and $2\; \backslash times\; x\; \backslash times\; y$ may be written $2xy$.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, [https://books.google.com/books?id=nL5ObMmDvPEC&dq=%22Algebraic+notation%22+multiplication+omitted&pg=PR9-IA8 page 68]

Usually terms with the highest power (exponent), are written on the left, for example, $x^2$ is written to the left of {{mvar|x}}. When a coefficient is one, it is usually omitted (e.g. $1x^2$ is written $x^2$).David Alan Herzog, Teach Yourself Visually Algebra, Publisher John Wiley & Sons, 2008, {{ISBN|0470185597}}, 9780470185599, 304 pages, [https://books.google.com/books?id=Igs6t_clf0oC&q=coefficient+of+1&pg=PA72 page 72] Likewise when the exponent (power) is one, (e.g. $3x^1$ is written $3x$).John C. Peterson, Technical Mathematics With Calculus, Publisher Cengage Learning, 2003, {{ISBN|0766861899}}, 9780766861893, 1613 pages, [https://books.google.com/books?id=PGuSDjHvircC&dq=%22when+the+exponent+is+1%22&pg=PA32 page 31] When the exponent is zero, the result is always 1 (e.g. $x^0$ is always rewritten to {{mvar|1}}).Jerome E. Kaufmann, Karen L. Schwitters, Algebra for College Students, Publisher Cengage Learning, 2010, {{ISBN|0538733543}}, 9780538733540, 803 pages, [https://books.google.com/books?id=-AHtC0IYMhYC&q=exponents+&pg=PA222 page 222] However $0^0$, being undefined, should not appear in an expression, and care should be taken in simplifying expressions in which variables may appear in exponents.

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. As an illustration of this, while exponents are usually formatted using superscripts, e.g., $x^2$, in plain text, and in the TeX mark-up language, the caret symbol {{char|^}} represents exponentiation, so $x^2$ is written as "x^2".Ramesh Bangia, Dictionary of Information Technology, Publisher Laxmi Publications, Ltd., 2010, {{ISBN|9380298153}}, 9789380298153, [https://books.google.com/books?id=zQa5I2sHPKEC&q=exponentiation+caret&pg=PA212 page 212]George Grätzer, First Steps in LaTeX, Publisher Springer, 1999, {{ISBN|0817641327}}, 9780817641320, [https://books.google.com/books?id=mLdg5ZdDKToC&q=subscripts+and+superscripts+caret&pg=PA17 page 17] This also applies to some programming languages such as Lua. In programming languages such as Ada,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, [https://books.google.com/books?id=694P3YtXh-0C&q=double+star+exponentiate&pg=PA12 page 13] Fortran,C. Xavier, Fortran 77 And Numerical Methods, Publisher New Age International, 1994, {{ISBN|812240670X}}, 9788122406702, [https://books.google.com/books?id=WYMgF9WFty0C&dq=fortran+asterisk+exponentiation&pg=PA20 page 20] Perl,Randal Schwartz, Brian Foy, Tom Phoenix, Learning Perl, Publisher O'Reilly Media, Inc., 2011, {{ISBN|1449313140}}, 9781449313142, [https://books.google.com/books?id=l2IwEuRjeNwC&q=double+asterisk+exponentiation&pg=PA24 page 24] PythonMatthew A. Telles, Python Power!: The Comprehensive Guide, Publisher Course Technology PTR, 2008, {{ISBN|1598631586}}, 9781598631586, [https://books.google.com/books?id=754knV_fyf8C&q=double+asterisk+exponentiation&pg=PA46 page 46] and Ruby,Kevin C. Baird, Ruby by Example: Concepts and Code, Publisher No Starch Press, 2007, {{ISBN|1593271484}}, 9781593271480, [https://books.google.com/books?id=kq2dBNdAl3IC&q=double+asterisk+exponentiation&pg=PA72 page 72] a double asterisk is used, so $x^2$ is written as "x**2". Many programming languages and calculators use a single asterisk to represent the multiplication symbol,William P. Berlinghoff, Fernando Q. Gouvêa, Math through the Ages: A Gentle History for Teachers and Others, Publisher MAA, 2004, {{ISBN|0883857367}}, 9780883857366, [https://books.google.com/books?id=JAXNVaPt7uQC&dq=calculator+asterisk+multiplication&pg=PA75 page 75] and it must be explicitly used, for example, $3x$ is written "3*x".

File:Pi-equals-circumference-over-diametre.svg, its circumference {{mvar|c}}, divided by its diameter {{mvar|d}}, is equal to the constant pi, $\backslash pi$ (approximately 3.14).]]

{{Main|Variable (mathematics)}}

Elementary algebra builds on and extends arithmeticThomas Sonnabend, Mathematics for Teachers: An Interactive Approach for Grades K-8, Publisher: Cengage Learning, 2009, {{ISBN|0495561665}}, 9780495561668, 759 pages, [https://books.google.com/books?id=gBa2GzyXCF8C&q=extends+arithmetic&pg=PR17 page xvii] by introducing letters called variables to represent general (non-specified) numbers. This is useful for several reasons.

1. Variables may represent numbers whose values are not yet known. For example, if the temperature of the current day, C, is 20 degrees higher than the temperature of the previous day, P, then the problem can be described algebraically as $C\; =\; P\; +\; 20$.Lewis Hirsch, Arthur Goodman, Understanding Elementary Algebra With Geometry: A Course for College Students, Publisher: Cengage Learning, 2005, {{ISBN|0534999727}}, 9780534999728, 654 pages, [https://books.google.com/books?id=jsT7kqZubvIC&dq=%22elementary+algebra%22+variables+unknown&pg=PA48 page 48]
2. Variables allow one to describe general problems,Lawrence S. Leff, College Algebra: Barron's Ez-101 Study Keys, Publisher: Barron's Educational Series, 2005, {{ISBN|0764129147}}, 9780764129148, 230 pages, [https://books.google.com/books?id=XesryURrNKAC&dq=algebra+variables+generalize&pg=PA2 page 2] without specifying the values of the quantities that are involved. For example, it can be stated specifically that 5 minutes is equivalent to $60\; \backslash times\; 5\; =\; 300$ seconds. A more general (algebraic) description may state that the number of seconds, $s\; =\; 60\; \backslash times\; m$, where m is the number of minutes.
3. Variables allow one to describe mathematical relationships between quantities that may vary.Ron Larson, Kimberly Nolting, Elementary Algebra, Publisher: Cengage Learning, 2009, {{ISBN|0547102275}}, 9780547102276, 622 pages, [https://books.google.com/books?id=U6v78M5nYKAC&q=relationships&pg=PA210 page 210] For example, the relationship between the circumference, c, and diameter, d, of a circle is described by $\backslash pi\; =\; c\; /d$.
4. 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 $(a\; +\; b)\; =\; (b\; +\; a)$.Charles P. McKeague, Elementary Algebra, Publisher: Cengage Learning, 2011, {{ISBN|0840064217}}, 9780840064219, 571 pages, [https://books.google.com/books?id=etTbP0rItQ4C&dq=%22elementary+algebra%22+commutative&pg=PA49 page 49]

{{Main|Expression (mathematics)|Computer algebra#Simplification}}

Algebraic expressions may be evaluated and simplified, based on the basic properties of arithmetic operations (addition, subtraction, multiplication, division and exponentiation). For example,

• Added terms are simplified using coefficients. For example, $x\; +\; x\; +\; x$ can be simplified as $3x$ (where 3 is a numerical coefficient).
• Multiplied terms are simplified using exponents. For example, $x\; \backslash times\; x\; \backslash times\; x$ is represented as $x^3$
• Like terms are added together,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, [https://books.google.com/books?id=o9GYQjZ7ZwUC&q=like+terms&pg=PA51 page 51] for example, $2x^2\; +\; 3ab\; -\; x^2\; +\; ab$ is written as $x^2\; +\; 4ab$, because the terms containing $x^2$ are added together, and the terms containing $ab$ are added together.
• Brackets can be "multiplied out", using the distributive property. For example, $x\; (2x\; +\; 3)$ can be written as $(x\; \backslash times\; 2x)\; +\; (x\; \backslash times\; 3)$ which can be written as $2x^2\; +\; 3x$
• Expressions can be factored. For example, $6x^5\; +\; 3x^2$, by dividing both terms by the common factor, $3x^2$ can be written as $3x^2\; (2x^3\; +\; 1)$

File:Pythagorean theorem - Ani.gif for a right-angle triangle, which shows the algebraic relationship between the triangle's hypotenuse, and the other two sides.]]

{{Main|Equation}}

An equation states that two expressions are equal using the symbol for equality, {{=}} (the equals sign).Mark Clark, Cynthia Anfinson, Beginning Algebra: Connecting Concepts Through Applications, Publisher Cengage Learning, 2011, {{ISBN|0534419380}}, 9780534419387, 793 pages, [https://books.google.com/books?id=wCzuRMC5048C&q=equation&pg=PA134 page 134] One of the best-known equations describes Pythagoras' law relating the length of the sides of a right angle triangle:Alan S. Tussy, R. David Gustafson, Elementary and Intermediate Algebra, Publisher Cengage Learning, 2012, {{ISBN|1111567689}}, 9781111567682, 1163 pages, [https://books.google.com/books?id=xqio_Xn4t7oC&dq=algebra+Pythagoras+hypotenuse&pg=PA493 page 493]

:$c^2\; =\; a^2\; +\; b^2$

This equation states that $c^2$, 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 {{mvar|a}} and {{mvar|b}}.

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 $a\; +\; b\; =\; b\; +\; a$); such equations are called identities. Conditional equations are true for only some values of the involved variables, e.g. $x^2\; -\; 1\; =\; 8$ is true only for $x\; =\; 3$ and $x\; =\; -3$. The values of the variables which make the equation true are the solutions of the equation and can be found through equation solving.

Another type of equation is 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: $a\; >\; b$ where $>$ represents 'greater than', and where 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.

By definition, equality is an equivalence relation, meaning it is reflexive (i.e. $b\; =\; b$), symmetric (i.e. if $a\; =\; b$ then $b\; =\; a$), and transitive (i.e. if $a\; =\; b$ and $b\; =\; c$ then $a\; =\; c$).Douglas Downing, Algebra the Easy Way, Publisher Barron's Educational Series, 2003, {{ISBN|0764119729}}, 9780764119729, 392 pages, [https://books.google.com/books?id=RiX-TJLiQv0C&dq=algebra+equality+++reflexive++symmetric++transitive&pg=PA20 page 20] 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:

• if $a\; =\; b$ and $c\; =\; d$ then $a\; +\; c\; =\; b\; +\; d$ and $ac\; =\; bd$;
• if $a\; =\; b$ then $a\; +\; c\; =\; b\; +\; c$ and $ac\; =\; bc$;
• more generally, for any function {{mvar|f}}, if $a=b$ then $f(a)\; =\; f(b)$.

The relations less than and greater than $>$ have the property of transitivity:Ron Larson, Robert Hostetler, Intermediate Algebra, Publisher Cengage Learning, 2008, {{ISBN|0618753524}}, 9780618753529, 857 pages, [https://books.google.com/books?id=b3vqad8tbiAC&dq=algebra+inequality+properties&pg=PA96 page 96]

• If     and     then   ;
• If     and     then   ;{{cite web|url=https://math.stackexchange.com/q/1043755 |title=What is the following property of inequality called? |date=November 29, 2014 |work=Stack Exchange |access-date=4 May 2018}}
• If     and   $c\; >\; 0$   then   ;
• If     and     then   .

By reversing the inequation, and $>$ can be swapped,Chris Carter, Physics: Facts and Practice for A Level, Publisher Oxford University Press, 2001, {{ISBN|019914768X}}, 9780199147687, 144 pages, [https://books.google.com/books?id=Ff9gxZPYafcC&q=turned+around&pg=PA50 page 50] for example:

• is equivalent to $b\; >\; a$

{{main|Substitution (algebra)}}

{{see also|Substitution (logic)}}

Substitution is replacing the terms in an expression to create a new expression. Substituting 3 for {{mvar|a}} in the expression {{math|a*5}} makes a new expression {{math|3*5}} with meaning {{math|15}}. Substituting the terms of a statement makes a new statement. When the original statement is true independently of the values of the terms, the statement created by substitutions is also true. Hence, definitions can be made in symbolic terms and interpreted through substitution: if $a^2:=a\backslash times\; a$ is meant as the definition of $a^2,$ as the product of {{mvar|a}} with itself, substituting {{math|3}} for {{mvar|a}} informs the reader of this statement that $3^2$ means {{math|1=3 × 3 = 9}}. Often it's not known whether the statement is true independently of the values of the terms. And, substitution allows one to derive restrictions on the possible values, or show what conditions the statement holds under. For example, taking the statement {{math|1=x + 1 = 0}}, if {{mvar|x}} is substituted with {{math|1}}, this implies {{math|1=1 + 1 = 2 = 0}}, which is false, which implies that if {{math|1=x + 1 = 0}} then {{mvar|x}} cannot be {{math|1}}.

If {{math|x}} and {{math|y}} are integers, rationals, or real numbers, then {{math|1=xy = 0}} implies {{math|1=x = 0}} or {{math|1=y = 0}}. Consider {{math|1=abc = 0}}. Then, substituting {{math|a}} for {{math|x}} and {{math|bc}} for {{math|y}}, we learn {{math|1=a = 0}} or {{math|1=bc = 0}}. Then we can substitute again, letting {{math|1=x = b}} and {{math|1=y = c}}, to show that if {{math|1=bc = 0}} then {{math|1=b = 0}} or {{math|1=c = 0}}. Therefore, if {{math|1=abc = 0}}, then {{math|1=a = 0}} or ({{math|1=b = 0}} or {{math|1=c = 0}}), so {{math|1=abc = 0}} implies {{math|1=a = 0}} or {{math|1=b = 0}} or {{math|1=c = 0}}.

If the original fact were stated as "{{math|1=ab = 0}} implies {{math|1=a = 0}} or {{math|1=b = 0}}", then when saying "consider {{math|1=abc = 0}}," we would have a conflict of terms when substituting. Yet the above logic is still valid to show that if {{math|1=abc = 0}} then {{math|1=a = 0}} or {{math|1=b = 0}} or {{math|1=c = 0}} if, instead of letting {{math|1=a = a}} and {{math|1=b = bc}}, one substitutes {{math|a}} for {{math|a}} and {{math|b}} for {{math|bc}} (and with {{math|1=bc = 0}}, substituting {{math|b}} for {{math|a}} and {{math|c}} for {{math|b}}). This shows that substituting for the terms in a statement isn't always the same as letting the terms from the statement equal the substituted terms. In this situation it's clear that if we substitute an expression {{math|a}} into the {{math|a}} term of the original equation, the {{math|a}} substituted does not refer to the {{math|a}} in the statement "{{math|1=ab = 0}} implies {{math|1=a = 0}} or {{math|1=b = 0}}."

{{see also|Equation solving}}

File:Algebraproblem.jpg

The following sections lay out examples of some of the types of algebraic equations that may be encountered.

{{Main|Linear equation}}

Linear equations are so-called, because when they are plotted, they describe a straight line. The simplest equations to solve are linear equations 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 the age of a child and add 4, the resulting answer is 12. How old is the child?

:Equivalent equation: $2x\; +\; 4\; =\; 12$ where {{mvar|x}} represent the child'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.{{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=[https://archive.org/details/allmathyoullever00slav/page/72 72]|url=https://archive.org/details/allmathyoullever00slav/page/72}} This problem and its solution are as follows:

File:Divide large.gif

In words: the child is 4 years old.

The general form of a linear equation with one variable, can be written as: $ax+b=c$

Following the same procedure (i.e. subtract {{mvar|b}} from both sides, and then divide by {{mvar|a}}), the general solution is given by $x=\backslash frac\{c-b\}\{a\}$

File:Linear-equations-two-unknowns.svg

A linear equation with two variables has many (i.e. an infinite number of) solutions.Sinha, The Pearson Guide to Quantitative Aptitude for CAT 2/ePublisher: Pearson Education India, 2010, {{ISBN|8131723666}}, 9788131723661, 599 pages, [https://books.google.com/books?id=eOnaFSKRSR0C&q=many+solutions&pg=PA195 page 195] For example:

:Problem in words: A father is 22 years older than his son. How old are they?

:Equivalent equation: $y\; =\; x\; +\; 22$ where {{mvar|y}} is the father's age, {{mvar|x}} is the son's age.

That cannot be worked out by itself. If the son's age was made known, then there would no longer be two unknowns (variables). The problem then 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 it was also revealed that:

; Problem in words

: In 10 years, the father will be twice as old as his son.

;Equivalent equation

: $\backslash begin\{align\}$

y + 10 &= 2 \times (x + 10)\\

y &= 2 \times (x + 10) - 10 && \text{Subtract 10 from both sides}\\

y &= 2x + 20 - 10 && \text{Multiple out brackets}\\

y &= 2x + 10 && \text{Simplify}

\end{align}

Now there are two related linear equations, each with two unknowns, which enables the production of a linear equation with just one variable, by subtracting one from the other (called the elimination method):Cynthia Y. Young, Precalculus, Publisher John Wiley & Sons, 2010, {{ISBN|0471756849}}, 9780471756842, 1175 pages, [https://books.google.com/books?id=9HRLAn326zEC&dq=linear+equation++two+variables++many+solutions&pg=PA699 page 699]

:$\backslash begin\{cases\}$

y = x + 22 & \text{First equation}\\

y = 2x + 10 & \text{Second equation}

\end{cases}

:$\backslash begin\{align\}$

&&&\text{Subtract the first equation from}\\

(y - y) &= (2x - x) +10 - 22 && \text{the second in order to remove } y\\

0 &= x - 12 && \text{Simplify}\\

12 &= x && \text{Add 12 to both sides}\\

x &= 12 && \text{Rearrange}

\end{align}

In other words, the son is aged 12, and since the father 22 years older, he must be 34. In 10 years, the son will be 22, and the father 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.

{{Main|Quadratic equation}}

[[File:Quadratic-equation.svg|thumb|right|Quadratic equation plot of $y\; =\; x^2\; +\; 3x\; -\; 10$ showing its roots at $x\; =\; -5$ and $x\; =\; 2$, and that the quadratic can be rewritten as $y\; =\; (x\; +\; 5)(x\; -\; 2)$

]]

A quadratic equation is one which includes a term with an exponent of 2, for example, $x^2$,Mary Jane Sterling, Algebra II For Dummies, Publisher: John Wiley & Sons, 2006, {{ISBN|0471775819}}, 9780471775812, 384 pages, [https://books.google.com/books?id=_0rTMuSpTY0C&dq=quadratic+equations&pg=PA37 page 37] and no term with higher exponent. The name derives from the Latin quadrus, meaning square.John T. Irwin, The Mystery to a Solution: Poe, Borges, and the Analytic Detective Story, Publisher JHU Press, 1996, {{ISBN|0801854660}}, 9780801854668, 512 pages, [https://books.google.com/books?id=jsxTenuOQKgC&dq=quadratic+quadrus&pg=PA372 page 372] In general, a quadratic equation can be expressed in the form $ax^2\; +\; bx\; +\; c\; =\; 0$,Sharma/khattar, The Pearson Guide To Objective Mathematics For Engineering Entrance Examinations, 3/E, Publisher Pearson Education India, 2010, {{ISBN|8131723631}}, 9788131723630, 1248 pages, [https://books.google.com/books?id=2v-f9x7-FlsC&dq=quadratic%20equations%20%20ax2%20%2B%20bx%20%2B%20c%20%3D%200&pg=RA13-PA33 page 621] where {{mvar|a}} 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 $ax^2$, which is known as the quadratic term. Hence $a\; \backslash neq\; 0$, and so we may divide by {{mvar|a}} and rearrange the equation into the standard form

: $x^2\; +\; px\; +\; q\; =\; 0$

where $p\; =\; \backslash frac\{b\}\{a\}$ and $q\; =\; \backslash frac\{c\}\{a\}$. Solving this, by a process known as completing the square, leads to the quadratic formula

:$x=\backslash frac\{-b\; \backslash pm\; \backslash sqrt\; \{b^2-4ac\}\}\{2a\},$

where the symbol "±" indicates that both

: $x=\backslash frac\{-b\; +\; \backslash sqrt\; \{b^2-4ac\}\}\{2a\}\backslash quad\backslash text\{and\}\backslash quad\; x=\backslash frac\{-b\; -\; \backslash sqrt\; \{b^2-4ac\}\}\{2a\}$

are solutions of the quadratic equation.

Quadratic equations can also be solved using factorization (the reverse process of which is expansion, but for two linear terms is sometimes denoted foiling). As an example of factoring:

: $x^\{2\}\; +\; 3x\; -\; 10\; =\; 0,$

which is the same thing as

: $(x\; +\; 5)(x\; -\; 2)\; =\; 0.$

It follows from the zero-product property that either $x\; =\; 2$ or $x\; =\; -5$ 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,

: $x^\{2\}\; +\; 1\; =\; 0$

has no real number solution since no real number squared equals −1.

Sometimes a quadratic equation has a root of multiplicity 2, such as:

: $(x\; +\; 1)^2\; =\; 0.$

For this equation, −1 is a root of multiplicity 2. This means −1 appears twice, since the equation can be rewritten in factored form as

:$[x-(-1)][x-(-1)]=0.$

All quadratic equations have exactly two solutions in complex numbers (but they may be equal to each other), a category that includes real numbers, imaginary numbers, and sums of real and imaginary numbers. Complex numbers first arise in the teaching of quadratic equations and the quadratic formula. For example, the quadratic equation

:$x^2+x+1=0$

has solutions

:$x=\backslash frac\{-1\; +\; \backslash sqrt\{-3\}\}\{2\}\; \backslash quad\; \backslash quad\; \backslash text\{and\}\; \backslash quad\; \backslash quad\; x=\backslash frac\{-1-\backslash sqrt\{-3\}\}\{2\}.$

Since $\backslash sqrt\{-3\}$ is not any real number, both of these solutions for x are complex numbers.

{{Main|Logarithm}}

File:Binary logarithm plot with ticks.svg of the logarithm to base 2 crosses the x axis (horizontal axis) at 1 and passes through the points with coordinates {{nowrap|(2, 1)}}, {{nowrap|(4, 2)}}, and {{nowrap|(8, 3)}}. For example, {{nowrap|log₂(8) {{=}} 3}}, because {{nowrap|2³ {{=}} 8.}} The graph gets arbitrarily close to the y axis, but does not meet or intersect it.]]

An exponential equation is one which has the form $a^x\; =\; b$ for $a\; >\; 0$,Aven Choo, LMAN OL Additional Maths Revision Guide 3, Publisher Pearson Education South Asia, 2007, {{ISBN|9810600011}}, 9789810600013, [https://books.google.com/books?id=NsBXDMrzcJIC&dq=%22+exponential+equation+%22+aX+%3D+b&pg=RA2-PA29 page 105] which has solution

: $x\; =\; \backslash log\_a\; b\; =\; \backslash frac\{\backslash ln\; b\}\{\backslash ln\; a\}$

when $b\; >\; 0$. Elementary algebraic techniques are used to rewrite a given equation in the above way before arriving at the solution. For example, if

: $3\; \backslash cdot\; 2^\{x\; -\; 1\}\; +\; 1\; =\; 10$

then, by subtracting 1 from both sides of the equation, and then dividing both sides by 3 we obtain

: $2^\{x\; -\; 1\}\; =\; 3$

whence

: $x\; -\; 1\; =\; \backslash log\_2\; 3$

or

: $x\; =\; \backslash log\_2\; 3\; +\; 1.$

A logarithmic equation is an equation of the form $log\_a(x)\; =\; b$ for $a\; >\; 0$, which has solution

: $x\; =\; a^b.$

For example, if

: $4\backslash log\_5(x\; -\; 3)\; -\; 2\; =\; 6$

then, by adding 2 to both sides of the equation, followed by dividing both sides by 4, we get

: $\backslash log\_5(x\; -\; 3)\; =\; 2$

whence

: $x\; -\; 3\; =\; 5^2\; =\; 25$

from which we obtain

: $x\; =\; 28.$

{{Image frame|align=right|width=150|caption=Radical equation showing two ways to represent the same expression. The triple bar means the equation is true for all values of x|content=$\backslash overset\{\}\{\backslash underset\{\}\{\backslash sqrt[2]\{x^3\}\; \backslash equiv\; x^\{\backslash frac\; 3\; 2\}\; \}\; \}$}}

A radical equation is one that includes a radical sign, which includes square roots, $\backslash sqrt\{x\},$ cube roots, $\backslash sqrt[3]\{x\}$, and nth roots, $\backslash sqrt[n]\{x\}$. Recall that an nth root can be rewritten in exponential format, so that $\backslash sqrt[n]\{x\}$ is equivalent to $x^\{\backslash frac\{1\}\{n\}\}$. Combined with regular exponents (powers), then $\backslash sqrt[2]\{x^3\}$ (the square root of {{mvar|x}} cubed), can be rewritten as $x^\{\backslash frac\{3\}\{2\}\}$.John C. Peterson, Technical Mathematics With Calculus, Publisher Cengage Learning, 2003, {{ISBN|0766861899}}, 9780766861893, 1613 pages, [https://books.google.com/books?id=PGuSDjHvircC&dq=%22+radical+equation%22&pg=PA525 page 525] So a common form of a radical equation is $\backslash sqrt[n]\{x^m\}=a$ (equivalent to $x^\backslash frac\{m\}\{n\}=a$) where {{mvar|m}} and {{mvar|n}} are integers. It has real solution(s):

For example, if:

:$(x\; +\; 5)^\{2/3\}\; =\; 4$

then

: $\backslash begin\{align\}$

x + 5 & = \pm (\sqrt{4})^3,\\

x + 5 & = \pm 8,\\

x & = -5 \pm 8,

\end{align}

and thus

:$x\; =\; 3\; \backslash quad\; \backslash text\{or\}\backslash quad\; x\; =\; -13$

{{Main|System of linear equations}}

There are different methods to solve a system of linear equations with two variables.

File:Intersecting Lines.svg

An example of solving a system of linear equations is by using the elimination method:

:

2x - y&= 1.\end{cases}

Multiplying the terms in the second equation by 2:

: $4x\; +\; 2y\; =\; 14$

: $4x\; -\; 2y\; =\; 2.$

Adding the two equations together to get:

: $8x\; =\; 16$

which simplifies to

: $x\; =\; 2.$

Since the fact that $x\; =\; 2$ is known, it is then possible to deduce that $y\; =\; 3$ by either of the original two equations (by using 2 instead of {{mvar|x}} ) The full solution to this problem is then

: $\backslash begin\{cases\}\; x\; =\; 2\; \backslash \backslash \; y\; =\; 3.\; \backslash end\{cases\}$

This is not the only way to solve this specific system; {{mvar|y}} could have been resolved before {{mvar|x}}.

Another way of solving the same system of linear equations is by substitution.

:

\\ 2x - y &= 1.\end{cases}

An equivalent for {{mvar|y}} can be deduced by using one of the two equations. Using the second equation:

: $2x\; -\; y\; =\; 1$

Subtracting $2x$ from each side of the equation:

:

- y & = 1 - 2x

\end{align}

and multiplying by −1:

: $y\; =\; 2x\; -\; 1.$

Using this {{mvar|y}} value in the first equation in the original system:

:

4x + 4x - 2 &= 14 \\

8x - 2 &= 14 \end{align}

Adding 2 on each side of the equation:

:

8x &= 16 \end{align}

which simplifies to

: $x\; =\; 2$

Using this value in one of the equations, the same solution as in the previous method is obtained.

: $\backslash begin\{cases\}\; x\; =\; 2\; \backslash \backslash \; y\; =\; 3.\; \backslash end\{cases\}$

This is not the only way to solve this specific system; in this case as well, {{mvar|y}} could have been solved before {{mvar|x}}.

File:Parallel Lines.svg

File:Quadratic-linear-equations.svg

In the above example, a solution exists. However, there are also systems of equations which do not have any solution. Such a system is called inconsistent. An obvious example is

:

0x + 0y &= 2\,. \end{align} \end{cases}

As 0≠2, the second equation in the system has no solution. Therefore, the system has no solution.

However, not all inconsistent systems are recognized at first sight. As an example, consider the system

:

-2x - y &= -4\,. \end{align}\end{cases}

Multiplying by 2 both sides of the second equation, and adding it to the first one results in

: $0x+0y\; =\; 4\; \backslash ,,$

which clearly has no 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 {{mvar|x}} and {{mvar|y}}) For example:

:

-2x - y & = -6 \end{align}\end{cases}

Isolating {{mvar|y}} in the second equation:

: $y\; =\; -2x\; +\; 6$

And using this value in the first equation in the system:

: $\backslash begin\{align\}4x\; +\; 2(-2x\; +\; 6)\; =\; 12\; \backslash \backslash$

4x - 4x + 12 = 12 \\

12 = 12 \end{align}

The equality is true, but it does not provide a value for {{mvar|x}}. Indeed, one can easily verify (by just filling in some values of {{mvar|x}}) that for any {{mvar|x}} there is a solution as long as $y\; =\; -2x\; +\; 6$. There is an infinite number of solutions for this system.

Systems with more variables than the number of linear equations are called underdetermined. Such a system, if it has any solutions, does not have a unique one but rather an infinitude of them. An example of such a system is

:

y - z & = 2 .\end{align}\end{cases}

When trying to solve it, one is led to express some variables as functions of the other ones if any solutions exist, but cannot express all solutions numerically because there are an infinite number of them if there are any.

A system with a higher number of equations than variables is called overdetermined. If an overdetermined system has any solutions, necessarily some equations are linear combinations of the others.

• History of algebra
• Binary operation
• Gaussian elimination
• Mathematics education
• Number line
• Polynomial
• Cancelling out
• Tarski's high school algebra problem

• Leonhard Euler, Elements of Algebra, 1770. English translation Tarquin Press, 2007, {{ISBN|978-1-899618-79-8}}, also online digitized editions[http://web.mat.bham.ac.uk/C.J.Sangwin/euler/ Euler's Elements of Algebra] {{webarchive|url=https://web.archive.org/web/20110413234352/http://web.mat.bham.ac.uk/C.J.Sangwin/euler/ |date=2011-04-13 }} 2006,{{cite web|url=https://books.google.com/books?id=X8yv0sj4_1YC&q=euler+elements|title=Elements of Algebra|first1=Leonhard|last1=Euler|first2=John|last2=Hewlett|first3=Francis|last3=Horner|first4=Jean|last4=Bernoulli|first5=Joseph Louis|last5=Lagrange|date=4 May 2018|publisher=Longman, Orme|access-date=4 May 2018|via=Google Books}} 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] {{Webarchive|url=https://web.archive.org/web/20160610165651/http://catalog.flatworldknowledge.com/bookhub/reader/128?e=fwk-redden-ch01 |date=2016-06-10 }}. Flat World Knowledge, 2011

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