Elementary algebra

The elementary algebra is the basic form of algebra. In contrast to the arithmetic occur in elementary algebra in addition to numbers and basic arithmetic operations and variables. In contrast to the abstract algebra no algebraic structures such as vector spaces are considered in elementary algebra.

Variables

The addition of variables to the numbers and basic arithmetic has the advantage that general laws can be formulated precisely and clearly above all. Basic laws of real numbers for example, the commutative, the associative law or the distributive.

In addition, you can set variables with equations or inequalities and to testing for solvability. An example of an equation with one variable. If the definition is set for the set of rational numbers, then this equation has exactly one solution, namely. Substituting these numbers into the equation for a, produces a true statement for all other insertions false statements. Allowed for only insertions with integers, then the equation has no solution.

The description of functional dependencies can be represented by means of variables: for example, they sold tickets at a price of 3 € and has fixed costs of 10 €, thus making a profit of €.

Terme

A term is clearly a useful mathematical sequence of characters. More precisely, there is a term in the algebra of numbers, variables, arithmetic operations ( including the four basic arithmetic operations, exponentiation, square root and the logarithm ), and staples as auxiliary characters.

One example. Contains a term variable, so he goes at replacing all variables by elements of the universal set into a number. It is important to note when dividing, that will not be divided by 0. When the square root may occur as radicands only non-negative numbers, and the logarithm as arguments only positive numbers.

As in arithmetic, it is also important in algebra, knowing exactly how mathematical terms are interpreted. This is from the rules of precedence of operations is determined ( for example, " point before dashes " brackets first calculate).

To solve equations and inequalities are required term transformations. For example, the expression can also be written as. These two terms are equivalent. The most important term transformations are obtained by application of the laws and rules of the numerical calculation. Such rules for generating equivalent terms are:

• The commutative and associative laws of addition and multiplication,
• The distributive ( clip rules)
• Binomial formulas
• The power laws as well as
• The Logarithmengesetze.

Equations and Inequalities

An equation consists of two terms, between which an equal sign. An inequality consists of two terms, between which an inequality sign is. Come in two terms, no variables before, then the (in) equation is a statement, otherwise a statement form. The amount of items that you may use for the variables is called ground set or set of definitions. Those elements of the definition amount at their establishment for the variables the (in) equation is a true statement, hot solutions of ( Un) equation. All solutions are summarizes the solution set.

For example, the equation is satisfied only for the values ​​2 and -2 of. The amount of solution therefore consists of the two elements -2 and 2, ie.

Some equations at each establishment from the definition amount to a true statement, for example. Such equations are called universal.

The most important method to solve equations ( inequalities ) are equivalence transformations. They do not change the solution set of the equation (inequality). Examples of equivalence transformations are:

• Replacing a term by an equivalent Term
• Adding or subtracting the same numbers (terms) on both sides of the equation (inequality).
• Multiplying or dividing both sides of equation ( inequality) with the same term, if this in any permissible setting takes the value 0. For inequalities, the " direction" of the inequality symbol must be reversed if the number that is multiplied or divided by, is negative.
• Logarithms, provided all terms only assume positive values ​​for all allowable substitutions. In inequalities may have to a case distinction for term values ​​greater and less than or equal values ​​for Term 1 are made.
• Are drawn from the standing either side of the equal sign terms the root, we obtain as an equivalent statement form the disjunction of two equations. The equation is equivalent to the disjunction.

No equivalence transformation, for example, squaring the root solving equations.

Equations that are considered in the basic algebra, for example:

• Linear equations,
• Systems of linear equations,
• Quadratic equations,
• Simple cubic equations,
• Biquadratic equations
• Fraction equations
• Root equations
• Simple exponential and Logarithmengleichungen
• And the associated inequalities.

The use of at least capable graphic calculator or even better from calculators to a computer algebra system expands the possibilities for solving equations or inequalities considerably. It is possible to show visual solution quantities and to eliminate complicated term transformations.

Relationships

A commodity costs net € 140. What it costs 19% of gross value added tax? The relationship between net price, gross price and VAT can be expressed in words as follows: The gross price obtained by adding to the net price, VAT (19 % of the net price) added. Expressed in word variable is the context: Gross price = net price 19 % of the net price. Even clearer is if you use letters: B = N 19 % of N. Or equivalently transformed B = 1.19 · N. This equation now describes all possible net prices N the relationship with the associated gross prices as