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Encyclopedia > Elementary algebra

Elementary algebra is a fundamental and relatively basic form of algebra taught to students who are presumed to have little or no formal knowledge of mathematics beyond arithmetic. While in arithmetic only numbers and their arithmetical operations (such as +, −, ×, ÷) occur, in algebra one also uses symbols (such as x and y, or a and b) to denote numbers. These are called variables. This is useful because: This article is about the branch of mathematics. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word αριθμός = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple daily counting to advanced science and business calculations. ... A number is an abstract idea used in counting and measuring. ... In computer science and mathematics, a variable (IPA pronunciation: ) (sometimes called a pronumeral) is a symbolic representation denoting a quantity or expression. ...

  • It allows the generalization of arithmetical equations (and inequalities) to be stated as laws (such as a + b = b + a for all a and b), and thus is the first step to the systematic study of the properties of the real number system.
  • It allows reference to numbers which are not known. In the context of a problem, a variable may represent a certain value of which is uncertain, but may be solved through the formulation and manipulation of equations.
  • It allows the exploration of mathematical relationships between quantities (such as "if you sell x tickets, then your profit will be 3x − 10 dollars").

These three are the main strands of elementary algebra, which should be distinguished from abstract algebra, a more advanced topic generally taught to college students. An equation is a mathematical statement, in symbols, that two things are the same (or equivalent). ... For the socioeconomic meaning, see social inequality. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ...


In elementary algebra, an "expression" may contain numbers, variables and arithmetical operations. These are usually written (by convention) with 'higher-power' terms on the left (see polynomial); a few examples are: An expression is a combination of numbers, operators, grouping symbols (such as brackets and parentheses) and/or free variables and bound variables arranged in a meaningful way which can be evaluated. ... In mathematics, a polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ...

x + 3,
y^{2} + 2x - 3,
z^{7} + a(b + x^{3}) + 42/y - pi.,

In more advanced algebra, an expression may also include elementary functions. In mathematics, several functions are important enough to deserve their own name. ...


An "equation" is the claim that two expressions are equal. Some equations are true for all values of the involved variables (such as a + b = b + a); such equations are called "identities". Other equations are true for only some values of the involved variables: x2 − 1 = 4. The values of the variables which make the equation true are called the "solutions" of the equation. An equation is a mathematical statement, in symbols, that two things are the same (or equivalent). ... In mathematics, the term identity has several important uses: An identity is an equality that remains true regardless of the values of any variables that appear within it, to distinguish it from an equality which is true under more particular conditions. ...

Contents

Laws of elementary algebra[1]

 a - b = a + (-b).
Example: if 5 + x = 3 then x = − 2.
 {a over b} = a left( {1 over b} right).
  • Exponentiation is not a commutative operation.
    • Therefore exponentiation has a pair of reverse operations: logarithm and exponentiation with fractional exponents (e.g. square roots).
      • Examples: if 3x = 10 then x = log310. If x2 = 10 then x = 101 / 2.
    • The square roots of negative numbers do not exist in the real number system. (See: complex number system)
  • Associative property of addition: (a + b) + c = a + (b + c).
  • Associative property of multiplication: (ab)c = a(bc).
  • Distributive property of multiplication with respect to addition: c(a + b) = ca + cb.
  • Distributive property of exponentiation with respect to multiplication: (ab)c = acbc.
  • How to combine exponents: abac = ab + c.
  • Power to a power property of exponents: (ab)c = abc.

3 + 2 = 5 with apples, a popular choice in textbooks[1] Addition is the mathematical operation of combining or adding two numbers to obtain an equal simple amount or total. ... Mathematical meaning A map or binary operation is said to be commutative when, for any x in A and any y in B . ... 5 - 2 = 3 (verbally, five minus two equals three) An example problem Subtraction is one of the four basic arithmetic operations; it is essentially the opposite of addition. ... A negative number is a number that is less than zero, such as −3. ... In mathematics, multiplication is an elementary arithmetic operation. ... In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ... The reciprocal function: y = 1/x. ... Exponentiation is a mathematical operation, written an, involving two numbers, the base a and the exponent n. ... Logarithms to various bases: is to base e, is to base 10, and is to base 1. ... In mathematics, a square root of a number x is a number r such that , or in words, a number r whose square (the result of multiplying the number by itself) is x. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In mathematics, associativity is a property that a binary operation can have. ... In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ...

Laws of equality

In mathematics, a binary relation R over a set X is transitive if it holds for all a, b, and c in X, that if a is related to b and b is related to c, then a is related to c. ... In mathematics, two mathematical objects are considered equal if they are precisely the same in every way. ... In set theory, a binary relation can have, among other properties, reflexivity or irreflexivity. ... In mathematics, a binary relation R over a set X is symmetric if it holds for all a and b in X that if a is related to b then b is related to a. ...

Other laws

  • If a = b and c = d then a + c = b + d.
    • If a = b then a + c = b + c for any c (addition property of equality).
  • If a = b and c = d then ac = bd.
    • If a = b then ac = bc for any c (multiplication property of equality).
  • If two symbols are equal, then one can be substituted for the other at will (substitution principle).
  • If a > b and b > c then a > c (transitivity of inequality).
  • If a > b then a + c > b + c for any c.
  • If a > b and c > 0 then ac > bc.
  • If a > b and c < 0 then ac < bc.

For the use of the < and > signs in punctuation, see Bracket. ...

Examples

Linear equations in one variable

Main article: Linear equation

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. For example: A linear equation is an equation in which each term is either a constant or the product of a constant times the first power of a variable. ... A linear equation is an equation in which each term is either a constant or the product of a constant times the first power of a variable. ...

2x + 4 = 12. ,

The central 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. For example, by subtracting 4 from both sides in the equation above:

2x + 4 - 4 = 12 - 4 ,

which simplifies to:

2x = 8. ,

Dividing both sides by 2:

frac{2x}{2} = frac{8}{2} ,

simplifies to the solution:

x = 4. ,

The general case,

ax+b=c,

follows the same format for the solution:

x=frac{c-b}{a}

Quadratic equations

Main article: Quadratic equation

Quadratic equations can be expressed in the form ax2 + bx + c = 0, where 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 ax2, which is known as the quadratic term. Hence a ≠ 0, and so we may divide by a and rearrange the equation into the standard form In mathematics, a quadratic equation is a polynomial equation of the second degree. ... In mathematics, a quadratic equation is a polynomial equation of the second degree. ... A linear equation is an equation in which each term is either a constant or the product of a constant times the first power of a variable. ...

x^2 + px = q,

where p = b/a and q = −c/a. Solving this, by a process known as completing the square, leads to the quadratic formula. In mathematics, a quadratic equation is a polynomial equation of the second degree. ...


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: ... In mathematics, an expansion of a product of sums expresses it as a sum of products by using the fact that multiplication distributes over addition. ... A linear function is a mathematical function term of the form: f(x) = m x + c where c is a constant. ... The FOIL rule, also sometimes known as the double distributive property, is commonly taught to students learning algebra as a mnemonic (memory device) to remember how to multiply two binomials (polynomial with two terms). ...

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, The zero-product propety states that if the product of two elements a and b is zero, then one of a or b must be zero. ... Look up zero in Wiktionary, the free dictionary. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

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: In mathematics, the multiplicity of a member of a multiset is how many memberships in the multiset it has. ...

(x + 1)^{2} = 0. ,

For this equation, −1 is a root of multiplicity 2.


System of linear equations

In the case of a system of linear equations, like, for instance, two equations in two variables, it is often possible to find the solutions of both variables that satisfy both equations. In mathematics and linear algebra, a system of linear equations is a set of linear equations such as A standard problem is to decide if any assignment of values for the unknowns can satisfy all three equations simultaneously, and to find such an assignment if it exists. ... In mathematics and linear algebra, a system of linear equations is a set of linear equations such as A standard problem is to decide if any assignment of values for the unknowns can satisfy all three equations simultaneously, and to find such an assignment if it exists. ...


First method of finding a solution

An example of a system of linear equations could be the following:

begin{cases}4x + 2y = 14  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 x) The full solution to this problem is then

begin{cases} x = 2  y = 3. end{cases},

Note that this is not the only way to solve this specific system; y could have been solved before x.


Second method of finding a solution

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

begin{cases}4x + 2y = 14  2x - y = 1.end{cases} ,

An equivalent for 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:

2x - 2x - y = 1 - 2x ,
- y = 1 - 2x ,

and multiplying by -1:

 y = 2x - 1. ,

Using this y value in the first equation in the original system:

4x + 2(2x - 1) = 14 ,
4x + 4x - 2 = 14 ,
8x - 2 = 14 ,

Adding 2 on each side of the equation:

8x - 2 + 2 = 14 + 2 ,
8x = 16 ,

which simplifies to

x = 2 ,

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

begin{cases} x = 2  y = 3. end{cases},

Note that this is not the only way to solve this specific system; in this case as well, y could have been solved before x.


Other types of Systems of Linear Equations

Unsolvable Systems

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:

begin{cases} x + y = 1  0x + 0y = 2 end{cases},

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:

begin{cases}4x + 2y = 12  -2x - y = -4 end{cases},

When trying to solve this (for example, by using the method of substitution above), the second equation, after adding − 2x on both sides and multiplying by −1, results in:

y = -2x + 4 ,

And using this value for y in the first equation:

4x + 2(-2x + 4) = 12 ,
4x - 4x + 8 = 12 ,
8 = 12 ,

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 y obtained in the second equation.


Undetermined Systems

There are also systems which have multiple or infinite solutions, in opposition to a system with a unique solution (meaning, two unique values for x and y) For example:

begin{cases}4x + 2y = 12  -2x - y = -6 end{cases},

Isolating y in the second equation:

y = -2x + 6 ,

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

4x + 2(-2x + 6) = 12 ,
4x - 4x + 12 = 12 ,
12 = 12 ,

The equality is true, but it does not provide a value for x. Indeed, one can easily verify (by just filling in some values of x) that for any x there is a solution as long as y = −2x + 6. There are infinite 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

begin{cases}x + 2y = 10y - z = 2end{cases}

Such a system is called underdetermined; when trying to find a solution, one or more variables can only be expressed in relation to the other variables, but cannot be determined numerically. Incidentally, a system with a greater number of equations than variables, in which necessarily some equations are sums or multiples of others, is called overdetermined. In mathematics and linear algebra, a system of linear equations is a set of linear equations such as 3x1 + 2x2 − x3 = 1 2x1 − 2x2 + 4x3 = −2 −x1 + ½x2 − x3 = 0. ... This article discusses the use of the word Number in Mathematics. ... In mathematics, a system of linear equations is considered overdetermined if there are more equations than unknowns. ...


Relation between Solvability and Multiplicity

Given any system of linear equations, there is a relation between multiplicity and solvability.
If one equation is a multiple of the other (or, more generally, a sum of multiples of the other equations), then the system of linear equations is undetermined, meaning that the system has infinitely many solutions. Example: Multiple is a comic book superhero in the Marvel Comics universe. ... Addition is one of the basic operations of arithmetic. ...

begin{cases}x + y = 2  2x + 2y = 4end{cases}

When the multiplicity is only partial (meaning that for example, only the left hand sides of the equations are multiples, while the right hand sides are not or not by the same number) then the system is unsolvable. For example, in

begin{cases}x + y = 2  4x + 4y = 1end{cases}

the second equation yields that x + y = 1/4 which is in contradiction with the first equation. Such a system is also called inconsistent in the language of linear algebra. When trying to solve a system of linear equations it is generally a good idea to check if one equation is a multiple of the other. If this is precisely so, the solution cannot be uniquely determined. If this is only partially so, the solution does not exist.
This, however, does not mean that the equations must be multiples of each other to have a solution, as shown in the sections above; in other words: multiplicity in a system of linear equations is not a necessary condition for solvability. Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ... In logic, the words necessary and sufficient describe relations that hold between propositions or states of affairs, if one is conditional on the other. ...


See also

In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ... In mathematics, Gaussian elimination (not to be confused with Gauss–Jordan elimination), named after Carl Friedrich Gauss, is an algorithm in linear algebra for determining the solutions of a system of linear equations, for determining the rank of a matrix, and for calculating the inverse of an invertible square matrix. ... Mathematics education is a term that refers both to the practice of teaching and learning mathematics, as well as to a field of scholarly research on this practice. ... A number line, invented by John Wallis, is a one-dimensional picture in which the integers are shown as specially-marked points evenly spaced on a line. ... In mathematics, a polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ...

References

  • Leonhard [Euler], Elements of Algebra, Tarquin Press, 2007
  • Charles Smith, A Treatise on Algebra, in Cornell University Library Historical Math Monographs.

Footnotes

  1. ^ Mirsky, Lawrence (1990) An Introduction to Linear Algebra Library of Congress. p.72-3. ISBN 0-486-66434-1.

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