Elementary algebra

[4] This use of variables entails use of algebraic notation and an understanding of the general rules of the operations introduced in arithmetic: addition, subtraction, multiplication, division, etc.

It is typically taught to secondary school students and at introductory college level in the United States,[5] and builds on their understanding of arithmetic.

Many quantitative relationships in science and mathematics are expressed as algebraic equations.

In mathematics, a basic algebraic operation is any one of the common operations of elementary algebra, which include addition, subtraction, multiplication, division, raising to a whole number power, and taking roots (fractional power).

[12] Algebraic operations work in the same way as arithmetic operations,[13] such as addition, subtraction, multiplication, division and exponentiation,[14] 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.

[15] Usually terms with the highest power (exponent), are written on the left, for example,

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.

, in plain text, and in the TeX mark-up language, the caret symbol ^ represents exponentiation, so

In programming languages such as Ada,[20] Fortran,[21] Perl,[22] Python[23] and Ruby,[24] a double asterisk is used, so

Many programming languages and calculators use a single asterisk to represent the multiplication symbol,[25] and it must be explicitly used, for example,

Elementary algebra builds on and extends arithmetic[26] by introducing letters called variables to represent general (non-specified) numbers.

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

Conditional equations are true for only some values of the involved variables, e.g.

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.

And, substitution allows one to derive restrictions on the possible values, or show what conditions the statement holds under.

If x and y are integers, rationals, or real numbers, then xy = 0 implies x = 0 or y = 0.

If the original fact were stated as "ab = 0 implies a = 0 or b = 0", then when saying "consider abc = 0," we would have a conflict of terms when substituting.

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

Linear equations are so-called, because when they are plotted, they describe a straight line.

For example, if it was also revealed that: 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):[39] In other words, the son is aged 12, and since the father 22 years older, he must be 34.

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

Elementary algebraic techniques are used to rewrite a given equation in the above way before arriving at the solution.

, which has solution For example, if then, by adding 2 to both sides of the equation, followed by dividing both sides by 4, we get whence from which we obtain A radical equation is one that includes a radical sign, which includes square roots,

by either of the original two equations (by using 2 instead of x ) The full solution to this problem is then This is not the only way to solve this specific system; y could have been resolved before x.

As an example, consider the system Multiplying by 2 both sides of the second equation, and adding it to the first one results in which clearly has no solution.

Systems with more variables than the number of linear equations are called underdetermined.

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.