Elementary algebra

The solution to the quadratic equation $ax^2+bx+c=0$ . Here the symbols $a,b,c,x$ all are variables that represent numbers.

Two-dimensional plot (red curve) of the algebraic equation $y = x^2 - x - 2$

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,^[1] algebra introduces quantities without fixed values, known as variables.^[2] This use of variables entails a 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. Most quantitative results in science and mathematics are expressed as algebraic equations.

Algebraic notation

Main article: Mathematical notation

Algebraic notation describes how algebra is written. It follows certain rules and conventions, and has its own terminology. For example, the expression $3x^2 - 2xy + c$ has the following components:

Algebraic equation notation.svg
1 : Exponent (power), 2 : Coefficient, 3 : term, 4 : operator, 5 : constant, $x, y$ : variables

A coefficient is a numerical value which 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.^[3] Letters represent variables and constants. By convention, letters at the beginning of the alphabet (e.g. $a, b, c$ ) are typically used to represent constants, and those toward the end of the alphabet (e.g. $x, y$ and $z$ ) are used to represent variables.^[4] They are usually written in italics.^[5]

Algebraic operations work in the same way as arithmetic operations,^[6] such as addition, subtraction, multiplication, division and exponentiation.^[7] 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, $3 \times x^2$ is written as $3x^2$ , and $2 \times x \times y$ may be written $2xy$ .^[8]

Usually terms with the highest power (exponent), are written on the left, for example, $x^2$ is written to the left of $x$ . When a coefficient is one, it is usually omitted (e.g. $1x^2$ is written $x^2$ ).^[9] Likewise when the exponent (power) is one, (e.g. $3x^1$ is written $3x$ ).^[10] When the exponent is zero, the result is always 1 (e.g. $x^0$ is always rewritten to $1$ ).^[11] However $0^0$ , 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

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. For example, exponents are usually formatted using superscripts, e.g. $x^2$ . In plain text, and in the TeX mark-up language, the caret symbol "^" represents exponents, so $x^2$ is written as "x^2".^[12]^[13] In programming languages such as Ada,^[14] Fortran,^[15] Perl,^[16] Python ^[17] and Ruby,^[18] a double asterisk is used, so $x^2$ is written as "x**2". Many programming languages and calculators use a single asterisk to represent the multiplication symbol,^[19] and it must be explicitly used, for example, $3x$ is written "3*x".

Concepts

Variables

Example of variables showing the relationship between a circle's diameter and its circumference. For any circle, its circumference $c$ , divided by its diameter $d$ , is equal to the constant pi, $\pi$ (approximately 3.14).

Main article: Variable (mathematics)

Elementary algebra builds on and extends arithmetic^[20] by introducing letters called variables to represent general (non-specified) numbers. This is useful for several reasons.

Variables may represent numbers whose values are not yet known. For example, if the temperature today, T, is 20 degrees higher than the temperature yesterday, Y, then the problem can be described algebraically as $T = Y + 20$ .^[21]
Variables allow to describe general problems,^[22] without specifying the values of the quantities that are involved. For example, it can be stated specifically that 5 minutes is equivalent to $60 \times 5 = 300$ seconds. A more general (algebraic) description may state that the number of seconds, $s = 60 \times m$ , where m is the number of minutes.
Variables allow to describe mathematical relationships between quantities that may vary.^[23] For example, the relationship between the circumference, c, and diameter, d, of a circle is described by $\pi = c /d$ .
Variables allow to describe some mathematical properties. For example, a basic property of addition is commutativity which states that the order of numbers being added together does not matter. Commutativity is stated algebraically as $(a + b) = (b + a)$ .^[24]

Evaluating expressions

Main article: Expression (mathematics)

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

Added terms are simplified using coefficients. For example $x + x + x$ can be simplified as $3x$ (where 3 is the coefficient).
Multiplied terms are simplified using exponents. For example $x \times x \times x$ is represented as $x^3$
Like terms are added together,^[25] for example, $2x^2 + 3ab - x^2 + ab$ is written as $x^2 + 4ab$ , because the terms containing $x^2$ are added together, and, the terms containing $ab$ are added together.
Brackets can be "multiplied out", using distributivity. For example, $x (2x + 3)$ can be written as $(x \times 2x) + (x \times 3)$ which can be written as $2x^2 + 3x$
Expressions can be factored. For example, $6x^5 + 3x^2$ , by dividing both terms by $3x^2$ can be written as $3x^2 (2x^3 + 1)$

Equations

Animation illustrating Pythagoras' rule for a right-angle triangle, which shows the algebraic relationship between the triangle's hypotenuse, and the other two sides.

Main article: Equation

An equation states that two expressions are equal using the symbol for equality, $=$ (the equals sign).^[26] One of the most well-known equations describes Pythagoras' law relating the length of the sides of a right angle triangle:^[27]

$c^2 = a^2 + b^2$

This equation states that $c^2$ , 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 square 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 $a + b = b + a$ ); such equations are called identities. Conditional equations are true for only some values of the involved variables (e.g. $x^2 - 1 = 8$ is true only for $x = 3$ and $x = -3$ . 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 an 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: $a > b$ where $>$ represents 'greater than', and $a < b$ 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.

Properties of equality

By definition, equality follow a number of "equivalence relations", including (a) reflexive (i.e. $b = b$ ), (b) symmetric (i.e. if $a = b$ then $b = a$ ) (c) transitive (i.e. if $a = b$ and $b = c$ then $a = c$ ).^[28] From which:

if $a = b$ and $c = d$ then $a + c = b + d$ and $ac = bd$ ;
if $a = b$ then $a + c = b + c$ ;
if two symbols are equal, then one can be substituted for the other.

Properties of inequality

The relations less than $<$ and greater than $>$ have the property of transitivity:^[29]

If $a < b$ and $b < c$ then $a < c$ ;
If $a < b$ and $c < d$ then $a + c < b + d$ ;
If $a < b$ and $c > 0$ then $ac < bc$ ;
If $a < b$ and $c < 0$ then $bc < ac$ .

Note that by reversing the equation, we can swap $<$ and $>$ ,^[30] for example:

$a < b$ is equivalent to $b > a$

Solving algebraic equations

Linear equations with one variable

Main article: Linear equation

Linear equations are so-called, because when they are plotted, they describe a straight line (hence linear). 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:

Problem in words: If you double my son's age and add 4, the resulting answer is 12. How old is my son?

Equivalent equation: $2x + 4 = 12$ where $x$ represent my son'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.^[31] This problem and its solution are as follows:

1. Equation to solve:	$2x + 4 = 12$
2. Subtract 4 from both sides:	$2x + 4 - 4 = 12 - 4$
3. This simplifies to:	$2x = 8$
4. Divide both sides by 2:	$\frac{2x}{2} = \frac{8}{2}$
5. Simplifies to the solution:	$x = 4$

The general form of a linear equation with one variable, can be written as: $ax+b=c\,$

Following the same procedure (i.e. subtract $b$ from both sides, and then divide by $a$ ), the general solution is given by $x=\frac{c-b}{a}$

Linear equations with two variables

Solving two linear equations with a unique solution at the point that they intersect.

A linear equation with two variables has many (i.e. an infinite number of) solutions.^[32] For example:

Problem in words: I am 22 years older than my son. How old are we?

Equivalent equation: $y = x + 22$ where $y$ is my age, $x$ is my son's age.

This can not be worked out by itself. If I told you my son's age, then there would no longer be two unknowns (variables), and the problem 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 I also revealed that:

Problem in words:	In 10 years time, I will be twice as old as my son.
Equivalent equation:	$y + 10 = 2 \times (x + 10)$
Subtract 10 from both sides:	$y = 2 \times (x + 10) - 10$
Multiple out brackets:	$y = 2x + 20 - 10$
Simplify:	$y = 2x + 10$

Now there are two related linear equations, each with two unknowns, which lets us produce a linear equation with just one variable, by subtracting one from the other (called the elimination method):^[33]

Second equation	$y = 2x + 10$
First equation	$y = x + 22$
Subtract the first equation from the second in order to remove $y$	$(y - y) = (2x - x) +10 - 22$
Simplify	$0 = x - 12$
Add 12 to both sides	$12 = x$
Rearrange	$x = 12$

In other words, my son is aged 12, and as I am 22 years older, I must be 34. In 10 years time, my son will 22, and I 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

Main article: Quadratic equation

Quadratic equation plot of $y = x^2 + 3x - 10$ showing its roots at $x = -5$ and $x = 2$ , and that the quadratic can be rewritten as $y = (x + 5)(x - 2)$

A quadratic equation is one which includes a term with an exponent of 2, for example, $x^2$ ,^[34] and no term with higher exponent. The name derives from the Latin quadrus, meaning square.^[35] In general, a quadratic equation can be expressed in the form $ax^2 + bx + c = 0$ ,^[36] 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 $ax^2$ , which is known as the quadratic term. Hence $a \neq 0$ , and so we may divide by $a$ and rearrange the equation into the standard form

$x^2 + px + q = 0 \,$

where $p = b/a$ and $q = c/a$ . Solving this, by a process known as completing the square, leads to the quadratic formula

$x=\frac{-b \pm \sqrt {b^2-4ac}}{2a},$

where the symbol "±" indicates that both

$x=\frac{-b + \sqrt {b^2-4ac}}{2a}\quad\text{and}\quad x=\frac{-b - \sqrt {b^2-4ac}}{2a}$

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:

$x^{2} + 3x - 10 = 0. \,$

Which is the same thing as

$(x + 5)(x - 2) = 0. \,$

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,

$x^{2} + 1 = 0 \,$

has no real number solution since no real number squared equals −1. Sometimes a quadratic equation has a root of multiplicity 2, such as:

$(x + 1)^2 = 0. \,$

For this equation, −1 is a root of multiplicity 2. This means −1 appears two times.

Exponential and logarithmic equations

Main article: Logarithm

Graph showing a logarithm curves, which crosses the x-axis where x is 1 and extend towards minus infinity along the y-axis.

The graph of the logarithm to base 2 crosses the x axis (horizontal axis) at 1 and passes through the points with coordinates (2, 1), (4, 2), and (8, 3). For example, log₂(8) = 3, because 2³ = 8. The graph gets arbitrarily close to the y axis, but does not meet or intersect it.

An exponential equation is one which has the form $a^x = b$ for $a > 0$ ,^[37] which has solution

$X = \log_a b = \frac{\ln b}{\ln a}$

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

$3 \cdot 2^{x - 1} + 1 = 10$

then, by subtracting 1 from both sides of the equation, and then dividing both sides by 3 we obtain

$2^{x - 1} = 3\,$

whence

$x - 1 = \log_2 3\,$

$x = \log_2 3 + 1.\,$

A logarithmic equation is an equation of the form $log_a(x) = b$ for $a > 0$ , which has solution

$X = a^b.\,$

For example, if

$4\log_5(x - 3) - 2 = 6\,$

then, by adding 2 to both sides of the equation, followed by dividing both sides by 4, we get

$\log_5(x - 3) = 2\,$

whence

$x - 3 = 5^2 = 25\,$

from which we obtain

$x = 28.\,$

Radical equations

Radical equation showing two ways to represent the same expression

A radical equation is one that includes a radical (square root) sign, $\sqrt{x}$ , which includes cube roots, $\sqrt[3]{x}$ and nth roots, $\sqrt[n]{x}$ . Recall that an nth root can be rewritten in exponential format, so that $\sqrt[n]{x}$ is equivalent to $x^{\frac{1}{n}}$ . Combined with regular exponents (powers), then $\sqrt[2]{x^3}$ (the square root of $x$ cubed), can be rewritten as $x^{\frac{3}{2}}$ .^[38] So the general form of radical equation is $a = \sqrt[n]{x^m}$ (equivalent to $a = x^\frac{m}{n}$ ) where $m$ and $n$ integers, and which has solution:

$m$ is odd	$m$ is even and $a \ge 0$
$x = \sqrt[m]{a^n}$ or $x = \left(\sqrt[m]a\right)^n$	$x = \pm \sqrt[m]{a^n}$ or $x = \pm \left(\sqrt[m]a\right)^n$

For example, if:

$(x + 5)^{2/3} = 4,\,$

then

$\begin{align}x + 5 & = \pm (\sqrt{4})^3\x + 5 & = \pm 8\x & = -5 \pm 8\x & = 3,-13\end{align}$ .

System of linear equations

Main article: System of linear equations

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

Elimination method

The solution set for the equations $x - y = -1$ and $3x + y = 9$ is the single point (2, 3).

An example of solving a system of linear equations is by using the elimination method:

$\begin{cases}4x + 2y&= 14 \2x - y&= 1.\end{cases} \,$

Multiplying the terms in the second equation by 2:

$4x + 2y = 14 \,$

$4x - 2y = 2. \,$

Adding the two equations together to get:

$8x = 16 \,$

which simplifies to

$x = 2. \,$

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

$\begin{cases} x = 2 \ y = 3. \end{cases}\,$

Note that this is not the only way to solve this specific system; $y$ could have been solved before $x$ .

Substitution method

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

$\begin{cases}4x + 2y &= 14\ 2x - y &= 1.\end{cases} \,$

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

$2x - y = 1 \,$

Subtracting $2x$ from each side of the equation:

$\begin{align}2x - 2x - y & = 1 - 2x \- y & = 1 - 2x\end{align}$

and multiplying by −1:

$y = 2x - 1. \,$

Using this $y$ value in the first equation in the original system:

$\begin{align}4x + 2(2x - 1) &= 14\4x + 4x - 2 &= 14 \8x - 2 &= 14 \end{align}$

Adding 2 on each side of the equation:

$\begin{align}8x - 2 + 2 &= 14 + 2 \8x &= 16 \end{align}$

which simplifies to

$x = 2 \,$

Using this value in one of the equations, the same solution as in the previous method is obtained.

$\begin{cases} x = 2 \ y = 3. \end{cases}\,$

Note that this is not the only way to solve this specific system; in this case as well, $y$ could have been solved before $x$ .

Other types of systems of linear equations

Unsolvable systems

The equations $3x + 2y = 6$ and $3x + 2y = 12$ are parallel and cannot intersect, and is unsolvable.

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:

$\begin{cases}\begin{align} x + y &= 1 \0x + 0y &= 2 \end{align} \end{cases}\,$

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:

$\begin{cases}\begin{align}4x + 2y &= 12 \-2x - y &= -4 \end{align}\end{cases}\,$

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:

$y = -2x + 4 \,$

And using this value for $y$ in the first equation:

$\begin{align}4x + 2(-2x + 4) &= 12 \4x - 4x + 8 &= 12 \8 &= 12 \end{align}$

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

Plot of a quadratic equation (red) and a linear equation (blue) that do not intersect, and consequently for which there is no common 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:

$\begin{cases}\begin{align}4x + 2y & = 12 \-2x - y & = -6 \end{align}\end{cases}\,$

Isolating $y$ in the second equation:

$y = -2x + 6 \,$

And using this value in the first equation in the system:

$\begin{align}4x + 2(-2x + 6) = 12 \4x - 4x + 12 = 12 \12 = 12 \end{align}$

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 is an infinite number of 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

$\begin{cases}\begin{align}x + 2y & = 10\y - z & = 2 \end{align}\end{cases}$

Such a system is called underdetermined; when trying to solve it, one is lead to express some variables as functions of the other ones, but cannot express all solutions numerically. Incidentally, a system with a greater number of equations than variables, in which necessarily some equations are linear combination of the others, is called overdetermined.

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 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:

$\begin{cases} \begin{align} x + y &= 2 \ 2x + 2y &= 4 \end{align}\end{cases}$

has solutions for $(x, y)$ 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

$\begin{cases}\begin{align}x + y & = 2 \4x + 4y &= 1 \end{align}\end{cases}$

the second equation yields that $x + y = \frac{1}{4}$ 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.

References

Leonhard Euler, Elements of Algebra, 1770. English translation Tarquin Press, 2007, ISBN 978-1-899618-79-8, also online digitized editions^[39] 2006,^[40] 1822.
Charles Smith, A Treatise on Algebra, in Cornell University Library Historical Math Monographs.
Redden, John. Elementary Algebra. Flat World Knowledge, 2011

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^ John C. Peterson, Technical Mathematics With Calculus, Publisher Cengage Learning, 2003, ISBN 0766861899, 9780766861893, 1613 pages, page 525
^ Euler's Elements of Algebra
^ Elements of algebra – Leonhard Euler, John Hewlett, Francis Horner, Jean Bernoulli, Joseph Louis Lagrange – Google Books

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