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Wikipedia / Elementary algebra

Elementary algebra, also known as high school algebra or college algebra, encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces numerical variables (quantities without fixed values). In arithmetic, operations (such as addition and subtraction) can only be performed on numbers. In algebra, operations can be performed on numbers, variables, and terms. Unlike abstract algebra, elementary algebra is not concerned with algebraic structures outside the realm of real and complex numbers.

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

Algebraic notation

Algebraic notation describes the rules and conventions for writing mathematical expressions 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 ) are used to represent variabless. They are usually printed in italics.

Algebraic operations work in the same way as arithmetic operations, 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.

Usually terms with the highest power (exponent is written to the left of ).). is always rewritten to )., being undefined, should not appear in an expression, and care should be taken in simplifying expressions in which variables may appear in exponents.

Alternative notation

, in plain text, and in the TeX mark-up language, the caret symbol is written as "x^2". This also applies to some programming languages such as Lua. In programming languages such as Ada, Fortran, Perl, Python and Ruby, is written as "x**2". Many programming languages and calculators use a single asterisk to represent the multiplication symbol, is written "3*x".

Concepts

Variables

Elementary algebra builds on and extends arithmetic by introducing letters called variables to represent general (non-specified) numbers. This is useful for several reasons.

  1. Variables may represent numbers whose values are not yet known.

  2. Variables allow one to describe general problems, without specifying the values of the quantities that are involved., where m is the number of minutes.

  3. Variables allow one to describe mathematical relationships between quantities that may vary. For example, the relationship between the circumference, c, and diameter, d.

  4. Variables allow one to describe some mathematical properties. For example, a basic property of addition is commutativity.

Simplifying expressions

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

  • (where 3 is a numerical coefficient).

  • Like terms are added together, are added together.

  • Brackets can be "multiplied out", using the distributive property

  • Expressions can be factored, by dividing both terms by the common factor

Equations

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:

, 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 and .

); such equations are called identities. The values of the variables which make the equation true are the solutions of the equation and can be found through equation solving.

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.

Properties of equality

By definition, equality is an equivalence relation, meaning it is reflexive), symmetric), and transitive). 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:

  • ;

  • ;

  • more generally, for any function .

Properties of inequality

The relations less than have the property of transitivity:

  • If and then ;

  • If and then ;

  • If and then ;

  • If and then .

can be swapped, for example:

Substitution

Substitution is replacing the terms in an expression to create a new expression. Substituting 3 for in the expression makes a new expression with meaning as the product of with itself, substituting for means . 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 , if is substituted with , this implies , which is false, which implies that if then cannot be .

If and are integers, rationals, or real numbers, then implies or . Consider . Then, substituting for and for , we learn or . Then we can substitute again, letting and , to show that if then or . Therefore, if , then or ( or ), so implies or or .

If the original fact were stated as " implies or ", then when saying "consider ," we would have a conflict of terms when substituting. Yet the above logic is still valid to show that if then or or if, instead of letting and , one substitutes for and for (and with , substituting for and for ). 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 into the term of the original equation, the substituted does not refer to the in the statement " implies or ."

Solving algebraic equations

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

Linear equations with one variable

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:

  1. Problem in words: If you double the age of a child and add 4, the resulting answer is 12. How old is the child?

    where represent the child's age

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: In words: the child is 4 years old.

Following the same procedure (i.e. subtract from both sides, and then divide by

Linear equations with two variables

A linear equation with two variables has many (i.e. an infinite number of) solutions. For example:

  1. Problem in words: A father is 22 years older than his son. How old are they?

  2. where is the father's age, is the son's age.

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:

  1. Problem in words

  2. In 10 years, the father will be twice as old as his son.

  3. Equivalent equation

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.

Quadratic equations

, and no term with higher exponent. The name derives from the Latin quadrus, meaning square., where , and so we may divide by and rearrange the equation into the standard form

. 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 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

Complex numbers

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

has solutions

is not any real number, both of these solutions for x are complex numbers.

Exponential and logarithmic equations

, which has solution

. 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

, 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 one that includes a radical sign, which includes square roots cube roots, and nth roots. Recall that an n (the square root of .) where and are integers. It has real solution(s):For example, if:

then

and thus

System of linear equations

There are different methods to solve a system of linear equations with two variables.

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

by either of the original two equations (by using 2 instead of ) The full solution to this problem is then

This is not the only way to solve this specific system; could have been resolved before .

Substitution method

Another way of solving the same system of linear equations is by substitution.

An equivalent for can be deduced by using one of the two equations. Using the second equation:

from each side of the equation:

and multiplying by −1:

Using this 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, could have been solved before .

Other types of systems of linear equations

Inconsistent systems

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.

Undetermined systems

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 and ) For example:

Isolating 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 . Indeed, one can easily verify (by just filling in some values of ) that for any . There is an infinite number of solutions for this system.

Over- and underdetermined systems

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.

See also

References