Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics. It is typically taught to secondary school students and builds on their understanding of arithmetic. Whereas arithmetic deals with specified numbers,algebra introduces quantities without fixed values, known as variables. This use of variables entails use of algebraic notation and an understanding of the general rules of the operators introduced in arithmetic. Unlike abstract algebra, elementary algebra is not concerned with algebraic structures outside the realm of real and complex numbers.
The use of variables to denote quantities allows general relationships between quantities to be formally and concisely expressed, and thus enables solving a broader scope of problems. Many quantitative relationships in science and mathematics are expressed as algebraic equations.
Algebraic notation describes the rules and conventions for writing mathematical expressions, as well as the terminology used for talking about parts of expressions. For example, the expression has the following components:
A coefficient is a numerical value, or letter representing a numerical constant, that multiplies a variable (the operator is omitted). A term is an addend or a summand, a group of coefficients, variables, constants and exponents that may be separated from the other terms by the plus and minus operators. ) are typically used to represent constants, and those toward the end of the alphabet (e.g. and z) are used to represent variables. They are usually written in italics.Letters represent variables and constants. By convention, letters at the beginning of the alphabet (e.g.
Algebraic operations work in the same way as arithmetic operations, is written as , and may be written .such as addition, subtraction, multiplication, division and exponentiation. 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. For example,
Usually terms with the highest power (exponent), are written on the left, for example, is written to the left of x. When a coefficient is one, it is usually omitted (e.g. is written ). Likewise when the exponent (power) is one, (e.g. is written ). When the exponent is zero, the result is always 1 (e.g. is always rewritten to 1). However , being undefined, should not appear in an expression, and care should be taken in simplifying expressions in which variables may appear in exponents.
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. As an illustration of this, while exponents are usually formatted using superscripts, e.g., , in plain text, and in the TeX mark-up language, the caret symbol "^" represents exponentiation, so is written as "x^2"., as well as some programming languages such as Lua. In programming languages such as Ada, Fortran, Perl, Python and Ruby, a double asterisk is used, so is written as "x**2". Many programming languages and calculators use a single asterisk to represent the multiplication symbol, and it must be explicitly used, for example, is written "3*x".
Elementary algebra builds on and extends arithmeticby introducing letters called variables to represent general (non-specified) numbers. This is useful for several reasons.
Algebraic expressions may be evaluated and simplified, based on the basic properties of arithmetic operations (addition, subtraction, multiplication, division and exponentiation). For example,
An equation states that two expressions are equal using the symbol for equality, = (the equals sign). One of the best-known equations describes Pythagoras' law relating the length of the sides of a right angle triangle:
This equation states that , 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 a and 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 ); such equations are called identities. Conditional equations are true for only some values of the involved variables, e.g. is true only for and . 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: where represents 'greater than', and where 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.
By definition, equality is an equivalence relation, meaning it has the properties (a) reflexive (i.e. ), (b) symmetric (i.e. if then ) (c) transitive (i.e. if and then ). 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:
The relations less than and greater than have the property of transitivity:
By reversing the inequation, and can be swapped, for example:
Substitution is replacing the terms in an expression to create a new expression. Substituting 3 for a in the expression a*5 makes a new expression 3*5 with meaning 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 is meant as the definition of as the product of a with itself, substituting 3 for a informs the reader of this statement that means 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 x + 1 = 0, if x is substituted with 1, this implies 1 + 1 = 2 = 0, which is false, which implies that if x + 1 = 0 then x cannot be 1.
If x and y are integers, rationals, or real numbers, then xy = 0 implies x = 0 or y = 0. Consider abc = 0. Then, substituting a for x and bc for y, we learn a = 0 or bc = 0. Then we can substitute again, letting x = b and y = c, to show that if bc = 0 then b = 0 or c = 0. Therefore, if abc = 0, then a = 0 or (b = 0 or c = 0), so abc = 0 implies a = 0 or b = 0 or c = 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. Yet the above logic is still valid to show that if abc = 0 then a = 0 or b = 0 or c = 0 if, instead of letting a = a and b = bc, one substitutes a for a and b for bc (and with bc = 0, substituting b for a and c for 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 a into the a term of the original equation, the a substituted does not refer to the a in the statement "ab = 0 implies a = 0 or b = 0."
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. 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. As an example, consider:
To solve this kind of equation, the 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.This problem and its solution are as follows:
|1. Equation to solve:|
|2. Subtract 4 from both sides:|
|3. This simplifies to:|
|4. Divide both sides by 2:|
|5. This simplifies to the solution:|
In words: the child is 4 years old.
The general form of a linear equation with one variable, can be written as:
Following the same procedure (i.e. subtract b from both sides, and then divide by a), the general solution is given by
A linear equation with two variables has many (i.e. an infinite number of) solutions.For example:
That cannot be worked out by itself. If the son's age was made known, then there would no longer be two unknowns (variables). The problem then becomes a linear equation with just one variable, that can be solved as described above.
To solve a linear equation with two variables (unknowns), requires two related equations. 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):
In other words, the son is aged 12, and since the father 22 years older, he must be 34. In 10 years, the son will be 22, and the father will be twice his age, 44. This problem is illustrated on the associated plot of the equations.
For other ways to solve this kind of equations, see below, System of linear equations .
A quadratic equation is one which includes a term with an exponent of 2, for example, , and no term with higher exponent. The name derives from the Latin quadrus, meaning square. In general, a quadratic equation can be expressed in the form , 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 , which is known as the quadratic term. Hence , and so we may divide by a and rearrange the equation into the standard form
where and . Solving this, by a process known as completing the square, leads to the quadratic formula
where the symbol "±" indicates that both
are solutions of the quadratic equation.
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:
which is the same thing as
It follows from the zero-product property that either or 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,
has no real number solution since no real number squared equals −1. Sometimes a quadratic equation has a root of multiplicity 2, such as:
For this equation, −1 is a root of multiplicity 2. This means −1 appears twice, since the equation can be rewritten in factored form as
All quadratic equations have exactly two solutions in complex numbers (but they may be equal to each other), a category that includes real numbers, imaginary numbers, and sums of real and imaginary numbers. Complex numbers first arise in the teaching of quadratic equations and the quadratic formula. For example, the quadratic equation
Since is not any real number, both of these solutions for x are complex numbers.
An exponential equation is one which has the form for , which has solution
when . Elementary algebraic techniques are used to rewrite a given equation in the above way before arriving at the solution. For example, if
then, by subtracting 1 from both sides of the equation, and then dividing both sides by 3 we obtain
A logarithmic equation is an equation of the form for , 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
from which we obtain
A radical equation is one that includes a radical sign, which includes square roots, cube roots, , and nth roots, . Recall that an nth root can be rewritten in exponential format, so that is equivalent to . Combined with regular exponents (powers), then (the square root of x cubed), can be rewritten as . So a common form of a radical equation is (equivalent to ) where m and n are integers. It has real solution(s):
|n is odd||n is even|
|nandm are even|
|nis even,mis odd, and|
|no real solution|
For example, if:
There are different methods to solve a system of linear equations with two variables.
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 is known, it is then possible to deduce that 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.
Another way of solving the same system of linear equations is by substitution.
An equivalent for y can be deduced by using one of the two equations. Using the second equation:
Subtracting 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.
This is not the only way to solve this specific system; in this case as well, y could have been solved before x.
In the above example, a solution exists. However, there are also systems of equations which do not have any solution. Such a system is called inconsistent. An obvious example is
As 0≠2, the second equation in the system has no solution. Therefore, the system has no solution. However, not all inconsistent systems are recognized at first sight. 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.
There are also systems which have infinitely many solutions, in contrast to a system with a unique solution (meaning, a unique pair of values for x and y) For example:
Isolating y in the second equation:
And using this value in the first equation in the system:
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 . There is an infinite number of solutions for this system.
Systems with more variables than the number of linear equations are called underdetermined. Such a system, if it has any solutions, does not have a unique one but rather an infinitude of them. An example of such a system is
When trying to solve it, one is led to express some variables as functions of the other ones if any solutions exist, but cannot express all solutions numerically because there are an infinite number of them if there are any.
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.
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.
In mathematics, a Diophantine equation is a polynomial equation, usually involving two or more unknowns, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a constant the sum of two or more monomials, each of degree one. An exponential Diophantine equation is one in which unknowns can appear in exponents.
In mathematics, an equation is a statement that asserts the equality of two expressions, which are connected by the equals sign "=". The word equation and its cognates in other languages may have subtly different meanings; for example, in French an équation is defined as containing one or more variables, while in English, any equality is an equation.
In algebra, a quadratic equation is any equation that can be rearranged in standard form as
In elementary algebra, the quadratic formula is a formula that provides the solution(s) to a quadratic equation. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as factoring, completing the square, graphing and others.
In mathematics, a system of linear equations is a collection of one or more linear equations involving the same set of variables. For example,
In algebra, a cubic equation in one variable is an equation of the form
In algebra, a quadratic function, a quadratic polynomial, a polynomial of degree 2, or simply a quadratic, is a polynomial function with one or more variables in which the highest-degree term is of the second degree.
In mathematics, an implicit equation is a relation of the form R(x1,…, xn) = 0, where R is a function of several variables. For example, the implicit equation of the unit circle is x2 + y2 − 1 = 0.
In mathematics, a quartic equation is one which can be expressed as a quartic function equaling zero. The general form of a quartic equation is
In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form
In algebra, a quartic function is a function of the form
In mathematics, separation of variables is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation.
In mathematics, an algebraic equation or polynomial equation is an equation of the form
In mathematics, to solve an equation is to find its solutions, which are the values that fulfill the condition stated by the equation, consisting generally of two expressions related by an equals sign. When seeking a solution, one or more variables are designated as unknowns. A solution is an assignment of values to the unknown variables that makes the equality in the equation true. In other words, a solution is a value or a collection of values such that, when substituted for the unknowns, the equation becomes an equality. A solution of an equation is often called a root of the equation, particularly but not only for polynomial equations. The set of all solutions of an equation is its solution set.
In applied mathematics, in particular the context of nonlinear system analysis, a phase plane is a visual display of certain characteristics of certain kinds of differential equations; a coordinate plane with axes being the values of the two state variables, say, or etc.. It is a two-dimensional case of the general n-dimensional phase space.
In mathematics, a change of variables is a basic technique used to simplify problems in which the original variables are replaced with functions of other variables. The intent is that when expressed in new variables, the problem may become simpler, or equivalent to a better understood problem.
In mathematics, an algebraic expression is an expression built up from integer constants, variables, and the algebraic operations. For example, 3x2 − 2xy + c is an algebraic expression. Since taking the square root is the same as raising to the power 1/2,
In mathematics, a quadratic equation is a polynomial equation of the second degree. The general form is
In algebra, casus irreducibilis is one of the cases that may arise in attempting to solve polynomials of degree 3 or higher with integer coefficients, to obtain roots that are expressed with radicals. It shows that many algebraic numbers are real-valued but cannot be expressed in radicals without introducing complex numbers. The most notable occurrence of casus irreducibilis is in the case of cubic polynomials that are irreducible over the rational numbers and have three real roots, which was proven by Pierre Wantzel in 1843. One can decide whether a given irreducible cubic polynomial is in casus irreducibilis using the discriminant Δ, via Cardano's formula. Let the cubic equation be given by