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one of the main branches of
mathematics, it concerns the study of structure, relation and quantity.
Algebra studies the effects of adding and multiplying numbers,
variables, and polynomials, along with their factorization and
determining their roots. In addition to working directly with numbers,
algebra also covers symbols, variables, and set elements. Addition and
multiplication are general operations, but their precise definitions
lead to structures such as groups, rings, and fields.
(from Arabic al-jebr meaning
"reunion of broken parts") is the branch of mathematics concerning the
study of the rules of operations and relations, and the constructions
and concepts arising from them, including terms, polynomials, equations
and algebraic structures.
introduces the concept of
variables representing numbers. Statements based on these variables are
manipulated using the rules of operations that apply to numbers, such as
addition. This can be done for a variety of reasons, including equation
solving. in which the properties of operations on the real number
system are recorded using symbols as "place holders" to denote constants
and variables, and the rules governing mathematical expressions and
equations involving these symbols are studied. This is usually taught at
school under the title algebra (or intermediate algebra and college
algebra in subsequent years). University-level courses in group theory
may also be called elementary algebra.
The relation of equality (=) is...
reflexive: b = b;
symmetric: if a = b then b = a;
transitive: if a = b and b = c then a = c.
The relation of equality (=) has the property...
that if a = b and c = d then a + c = b + d and ac = bd;
that if a = b then a + c = b + c;
that if two symbols are equal, then one can be substituted for the other.
is to 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.
Elementary algebraic techniques
are used to rewrite a given equation in the above way before arriving at
the solution. then, by subtracting 1 from both sides of the equation,
and then dividing both sides by 3 we obtain
An example of solving a system of
linear equations is by using the elimination method: Multiplying the
terms in the second equation by 2: Adding the two equations together to
get: which simplifies to 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 Note
that this is not the only way to solve this specific system; y could
have been solved before x.
An equivalent for y can be
deduced by using one of the two equations. Using the second equation:
Subtracting 2x from each side of the equation: and multiplying by -1:
Using this y value in the first equation in the original system: Adding 2
on each side of the equation: which simplifies to Using this value in
one of the equations, the same solution as in the previous method is
obtained. 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.
Any real number can be added to both sides.
Any real number can be subtracted from both sides.
Any real number can be multiplied to both sides.
Any non-zero real number can divide both sides.
Some functions can be applied to both sides.
If an equation in algebra is known to be true, the following operations may be used to produce another true equation:
a value that represents a
quantity along a continuum, such as -5 (an integer), 4/3 (a rational
number that is not an integer), 8.6 (a rational number given by a finite
decimal representation), √2 (the square root of two, an algebraic
number that is not rational) and π (3.1415926535..., a transcendental
is a way of solving a functional
equation of two polynomials for a number of unknown parameters. It
relies on the fact that two polynomials are identical precisely when all
corresponding coefficients are equal. The method is used to bring
formulas into a desired form.
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