Elementary algebra
Encyclopedia
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
. It is typically taught in secondary school
under the term algebra. The major difference between algebra and arithmetic is the inclusion of variables
. While in arithmetic only number
s and their arithmetical operations (such as +, −, ×, ÷) occur, in algebra, one also uses variables such as x and y, or a and b to replace numbers.
); a few examples are:
In more advanced algebra, an expression may also include elementary functions.
A common mnemonic device for remembering this order is PEMDAS. Generally in Elementary Algebra, the use of brackets (often called parentheses) and their simple applications will be taught at most of the school
s in the world.
. Conditional 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 the solutions of the equation and can be found through equation solving
.
s that have only one variable. They contain only constant numbers and a single variable without an exponent. For example:
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:
can simplify to:
Dividing both sides by 2:
simplifies to the solution:
The general case,
follows the same procedure to obtain the solution:
s 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
where p = b/a and q = −c/a. Solving this, by a process known as completing the square
, leads to the quadratic formula.
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 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,
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 two times.
when b > 0. 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
whence
or
A logarithmic equation is an equation of the form logaX = b for a > 0, 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
s, which has solution
if m is odd, and solution, and
if m is even and a ≥ 0.
For example, if --------------------
then
or
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.
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:
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:
And using this value for y in the first equation:
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.
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 y = −2x + 6. There are infinite solutions for this system.
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
.
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:
has solutions (x,y) such as (1,1), (0,2), (1.8,0.2), (4,−2), (−3000.75,3002.75), and so on.
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
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.
Algebra
Algebra 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...
taught to students who are presumed to have little or no formal knowledge of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
beyond arithmetic
Arithmetic
Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...
. It is typically taught in secondary school
Secondary school
Secondary school is a term used to describe an educational institution where the final stage of schooling, known as secondary education and usually compulsory up to a specified age, takes place...
under the term algebra. The major difference between algebra and arithmetic is the inclusion of variables
Variable (mathematics)
In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...
. While in arithmetic only number
Number
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
s and their arithmetical operations (such as +, −, ×, ÷) occur, in algebra, one also uses variables such as x and y, or a and b to replace numbers.
Variables
The purpose of using variables, symbols that denote numbers, is to allow the making of generalizations in mathematics. This is useful because:- It allows arithmetical equationEquationAn equation is a mathematical statement that asserts the equality of two expressions. In modern notation, this is written by placing the expressions on either side of an equals sign , for examplex + 3 = 5\,asserts that x+3 is equal to 5...
s (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 systemReal numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
. - It allows reference to numbers which are not known. In the context of a problem, a variable may represent a certain value which is not yet known, but which may be found 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").
Expressions
In elementary algebra, an expression may contain numbers, variables and arithmetical operations. These are conventionally written with 'higher-power' terms on the left (see polynomialPolynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
); a few examples are:
In more advanced algebra, an expression may also include elementary functions.
Properties of operations
| Operation | Is Written | commutative | associative | identity element Identity element In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them... |
inverse operation |
|---|---|---|---|---|---|
| Addition | a + b | a + b = b + a | (a + b) + c = a + (b + c) | 0, which preserves numbers: a + 0 = a | Subtraction ( - ) |
| Multiplication | a × b or a•b | a × b = b × a | (a × b) × c = a × (b × c) | 1, which preserves numbers: a × 1 = a | Division ( / ) |
| Exponentiation | ab or a^b | Not commutative ab≠ba | Not associative | 1, which preserves numbers: a1 = a | Logarithm (Log) |
- The operation of additionAdditionAddition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples....
...- means repeated addition of ones: n = 1 + 1 +...+ 1 (n number of times);
- has an inverse operation called subtraction: (a + b) − b = a, which is the same as adding a negative number, a − b = a + (−b);
- The operation of multiplicationMultiplicationMultiplication is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic ....
...- means repeated addition: a × n = a + a +...+ a (n number of times);
- has an inverse operation called division which is defined for non-zero numbers: (ab)/b = a, which is the same as multiplying by a reciprocal, a/b = a(1/b);
- distributesDistributivityIn mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalizes the distributive law from elementary algebra.For example:...
over addition: (a + b)c = ac + bc; - is abbreviated by juxtaposition: a × b ≡ ab;
- The operation of exponentiationExponentiationExponentiation is a mathematical operation, written as an, involving two numbers, the base a and the exponent n...
...- means repeated multiplication: an = a × a ×...× a (n number of times);
- has an inverse operation, called the logarithmLogarithmThe logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...
: alogab = b = logaab; - distributes over multiplication: (ab)c = acbc;
- can be written in terms of n-th roots: am/n ≡ (n√a)m and thus even roots of negative numbers do not exist in the real number system. (See: complex number systemComplex numberA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
) - has the property: abac = ab + c;
- has the property: (ab)c = abc.
- in general ab ≠ ba and (ab)c ≠ a(bc);
Order of operations
In mathematics it is important that the value of an expression is always computed the same way. Therefore, it is necessary to compute the parts of an expression in a particular order, known as the order of operations. The standard order of operations is expressed in the following chart.-
-
- parenthesis and other grouping symbols including brackets, absolute value symbols, and the fraction bar
- exponents and roots
- multiplication and division
- addition and subtraction
-
A common mnemonic device for remembering this order is PEMDAS. Generally in Elementary Algebra, the use of brackets (often called parentheses) and their simple applications will be taught at most of the school
School
A school is an institution designed for the teaching of students under the direction of teachers. Most countries have systems of formal education, which is commonly compulsory. In these systems, students progress through a series of schools...
s in the world.
Equations
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 a + b = b + a); such equations are called identitiesIdentity (mathematics)
In mathematics, the term identity has several different important meanings:*An identity is a relation which is tautologically true. This means that whatever the number or value may be, the answer stays the same. For example, algebraically, this occurs if an equation is satisfied for all values of...
. Conditional 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 the solutions of the equation and can be found through equation solving
Equation solving
In mathematics, to solve an equation is to find what values fulfill a condition stated in the form of an equation . These expressions contain one or more unknowns, which are free variables for which values are sought that cause the condition to be fulfilled...
.
Properties of equality
- The relation of equality (=) is...
- reflexiveReflexive relationIn mathematics, a reflexive relation is a binary relation on a set for which every element is related to itself, i.e., a relation ~ on S where x~x holds true for every x in S. For example, ~ could be "is equal to".-Related terms:...
: b = b; - symmetricSymmetric relationIn 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.In mathematical notation, this is:...
: if a = b then b = a; - transitiveTransitive relationIn mathematics, a binary relation R over a set X is transitive if whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c....
: if a = b and b = c then a = c.
- reflexive
- 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.
Properties of inequality
- The relation of inequality (<) has the property...
- of transivity: if a < b and b < c then a < c;
- that if a < b and c < d then a + c < b + d;
- that if a < b and c > 0 then ac < bc;
- that if a < b and c < 0 then bc < ac.
Algebraic examples
The following sections lay out examples of some of the types of alegbraic equations you might encounter.Linear equations in one variable
The simplest equations to solve are linear equationLinear equation
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable....
s that have only one variable. They contain only constant numbers and a single variable without an exponent. For example:
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:
can simplify to:
Dividing both sides by 2:
simplifies to the solution:
The general case,
follows the same procedure to obtain the solution:
Quadratic equations
Quadratic equationQuadratic equation
In mathematics, a quadratic equation is a univariate polynomial equation of the second degree. A general quadratic equation can be written in the formax^2+bx+c=0,\,...
s 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
where p = b/a and q = −c/a. Solving this, by a process known as completing the square
Completing the square
In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the formax^2 + bx + c\,\!to the formIn this context, "constant" means not depending on x. The expression inside the parenthesis is of the form ...
, leads to the quadratic formula.
Quadratic equations can also be solved using factorization
Factorization
In mathematics, factorization or factoring is the decomposition of an object into a product of other objects, or factors, which when multiplied together give the original...
(the reverse process of which is expansion
Polynomial expansion
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...
, but for two linear terms
Linear function
In mathematics, the term linear function can refer to either of two different but related concepts:* a first-degree polynomial function of one variable;* a map between two vector spaces that preserves vector addition and scalar multiplication....
is sometimes denoted foiling
FOIL rule
In elementary algebra, FOIL is a mnemonic for the standard method of multiplying two binomials—hence the method may be referred to as the FOIL method...
). As an example of factoring:
Which is the same thing as
It follows from the zero-product property
Zero-product property
In the mathematical areas of algebra and analysis, the zero-product property, generally known as the nonexistence of zero divisors, and also called the zero-product rule, the rule of zero product, or any other similar name, is an abstract and explicit statement of the familiar property from...
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
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
system, but need not have any in the real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
system. For example,
has no real number solution since no real number squared equals −1.
Sometimes a quadratic equation has a root of multiplicity
Multiplicity (mathematics)
In mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset. For example, the number of times a given polynomial equation has a root at a given point....
2, such as:
For this equation, −1 is a root of multiplicity 2. This means -1 appears two times.
Exponential and logarithmic equations
An exponential equation is an equation of the form aX = b for a > 0, which has solution
when b > 0. 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
whence
or
A logarithmic equation is an equation of the form logaX = b for a > 0, 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
Radical equations
A radical equation is an equation of the form Xm/n = a, for m, n integerInteger
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s, which has solution
if m is odd, and solution, and
if m is even and a ≥ 0.
For example, if --------------------
then
or
-
.
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.Elimination Method
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.
Second method of finding a solution
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 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.
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:
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:
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:
And using this value for y in the first equation:
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:
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 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
Such a system is called underdetermined
Underdetermined system
In mathematics, a system of linear equations is considered underdetermined if there are fewer equations than unknowns. The terminology can be described in terms of the concept of counting constraints. Each unknown can be seen as an available degree of freedom...
; 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
Number
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
.
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
Overdetermined system
In mathematics, a system of linear equations is considered overdetermined if there are more equations than unknowns. The terminology can be described in terms of the concept of counting constraints. Each unknown can be seen as an available degree of freedom...
.
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
Multiple (mathematics)
In mathematics, a multiple is the product of any quantity and an integer. In other words, for the quantities a and b, we say that b is a multiple of a if b = na for some integer n , which is called the multiplier or coefficient. If a is not zero, this is equivalent to saying that b/a is an integer...
of the other (or, more generally, a sum
SUM
SUM can refer to:* The State University of Management* Soccer United Marketing* Society for the Establishment of Useful Manufactures* StartUp-Manager* Software User’s Manual,as from DOD-STD-2 167A, and MIL-STD-498...
of multiples of the other equations), then the system of linear equations is undetermined, meaning that the system has infinitely many solutions. Example:
has solutions (x,y) such as (1,1), (0,2), (1.8,0.2), (4,−2), (−3000.75,3002.75), and so on.
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
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
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
.
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.
See also
- Binary operationBinary operationIn mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division....
- Gaussian eliminationGaussian eliminationIn linear algebra, Gaussian elimination is an algorithm for solving systems of linear equations. It can also be used to find the rank of a matrix, to calculate the determinant of a matrix, and to calculate the inverse of an invertible square matrix...
- Mathematics educationMathematics educationIn contemporary education, mathematics education is the practice of teaching and learning mathematics, along with the associated scholarly research....
- Number lineNumber lineIn basic mathematics, a number line is a picture of a straight line on which every point is assumed to correspond to a real number and every real number to a point. Often the integers are shown as specially-marked points evenly spaced on the line...
- PolynomialPolynomialIn mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...