Editing Elementary algebra (section)

==Concepts==

===Variables===
[[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]] {{mvar|c}}, divided by its [[diameter]] {{mvar|d}}, is equal to the constant [[pi]], <math>\pi</math> (approximately 3.14).]]
{{Main|Variable (mathematics)}}
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, [https://books.google.com/books?id=gBa2GzyXCF8C&q=extends+arithmetic&pg=PR17 page xvii]</ref> by introducing letters called variables to represent general (non-specified) numbers.  This is useful for several reasons.

#'''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 <math>C = P + 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, [https://books.google.com/books?id=jsT7kqZubvIC&dq=%22elementary+algebra%22+variables+unknown&pg=PA48 page 48]</ref>
#'''Variables allow one to describe ''general'' problems,<ref name=leff>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]</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.
#'''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, [https://books.google.com/books?id=U6v78M5nYKAC&q=relationships&pg=PA210 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>.
#'''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, [https://books.google.com/books?id=etTbP0rItQ4C&dq=%22elementary+algebra%22+commutative&pg=PA49 page 49]</ref>

=== Simplifying expressions ===
{{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 (mathematics)|division]] and [[exponentiation]]). For example,
*Added terms are simplified using coefficients. For example, <math>x + x + x</math> can be simplified as <math>3x</math> (where 3 is a numerical coefficient).
*Multiplied terms are simplified using exponents. For example, <math>x \times x \times x</math> is represented as <math>x^3</math>
*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, [https://books.google.com/books?id=o9GYQjZ7ZwUC&q=like+terms&pg=PA51 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.
*Brackets can be "multiplied out", using [[Distributive property|the distributive property]]. 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>
*Expressions can be [[Factorization|factored]]. For example, <math>6x^5 + 3x^2</math>, by dividing both terms by the common [[Factor (arithmetic)|factor]], <math>3x^2</math> can be written as <math>3x^2 (2x^3 + 1)</math>

=== Equations ===
[[File:Pythagorean theorem - Ani.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.]]
{{Main|Equation}}
An equation states that two expressions are equal using the symbol for equality, {{=}} (the [[equals sign]]).<ref>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]</ref> One of the best-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, [https://books.google.com/books?id=xqio_Xn4t7oC&dq=algebra+Pythagoras+hypotenuse&pg=PA493 page 493]</ref>

:<math>c^2 = a^2 + b^2</math>

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 {{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 <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]].

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: <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.

==== Properties of equality ====

By definition, equality is an [[equivalence relation]], meaning it is [[reflexive relation|reflexive]] (i.e. <math>b = b</math>), [[symmetric relation|symmetric]] (i.e. if <math>a = b</math> then <math>b = a</math>), and [[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, [https://books.google.com/books?id=RiX-TJLiQv0C&dq=algebra+equality+++reflexive++symmetric++transitive&pg=PA20 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:

* if <math>a = b</math> and <math>c = d</math> then <math>a + c = b + d</math> and <math>ac = bd</math>;
* if <math>a = b</math> then <math>a + c = b + c</math> and <math>ac = bc</math>;
* more generally, for any function {{mvar|f}}, if <math>a=b</math> then <math>f(a) = f(b)</math>.

==== Properties of inequality ====

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, [https://books.google.com/books?id=b3vqad8tbiAC&dq=algebra+inequality+properties&pg=PA96 page 96]</ref>
* If &nbsp; <math>a < b</math> &nbsp; and &nbsp; <math>b < c</math> &nbsp; then &nbsp; <math>a < c</math>;
* If &nbsp; <math>a < b</math> &nbsp; and &nbsp; <math>c < d</math> &nbsp; then &nbsp; <math>a + c < b + d</math>;<ref>{{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}}</ref>
* If &nbsp; <math>a < b</math> &nbsp; and &nbsp; <math>c > 0</math> &nbsp; then &nbsp; <math>ac < bc</math>;
* If &nbsp; <math>a < b</math> &nbsp; and &nbsp; <math>c < 0</math> &nbsp; then &nbsp; <math>bc < ac</math>.
By reversing the inequation, <math> < </math> and <math> > </math> can be swapped,<ref>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]</ref> for example:
* <math>a < b</math> is equivalent to <math>b > a</math>

=== Substitution ===
{{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 <math>a^2:=a\times a</math> is meant as the definition of <math>a^2,</math> as the product of {{mvar|a}} with itself, substituting {{math|3}} for {{mvar|a}} informs the reader of this statement that <math>3^2</math> 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}}."