This article is about basic algebra in mathematics. For other uses of the term "algebra" see algebra (disambiguation).

Elementary algebra is the most basic form of algebra taught to students who are presumed to have no knowledge of mathematics beyond the basic principles of arithmetic. While in arithmetic only numbers and their arithmetical operations (such as +, −, ×, ÷) occur, in algebra one also uses symbols (such as a, x, y) to denote numbers. This is useful because:

• It allows the general formulation of arithmetical laws (such as $a\; +\; b\; =\; b\; +\; a$ for all a and b), and thus is the first step to a systematic exploration of the properties of the real number system.
• It allows the reference to "unknown" numbers, the formulation of equations and the study of how to solve these (for instance "find a number x such that $3x\; +\; 1\; =\; 10$").
• It allows the formulation of functional relationships (such as "if you sell x tickets, then your profit will be $3x\; -\; 10$ dollars").

These three are the main strands of elementary algebra, which should be distinguished from abstract algebra, a much more advanced topic generally taught to college seniors.

In algebra, an "expression" may contain numbers, variables and arithmetical operations. These are usually written (by convention) with higher-power terms on the left (see polynomial); a few examples are:

$x\; +\; 3\backslash ,$

$y^\{2\}\; +\; 2x\; -\; 3\backslash ,$

$z^\{7\}\; +\; a(b\; +\; x^\{3\})\; +\; 42/y\; -\; \backslash pi.\backslash ,$

An "equation" is the claim that two expressions are equal. Some equations are true for all values of the involved variables (such as $a\; +\; (b\; +\; c)\; =\; (a\; +\; b)\; +\; c$); these are also known as "identities". Other equations contain symbols for unknown values and we are then interested in finding those values for which the equation becomes true: $x^\{2\}\; -\; 1\; =\; 4.$ These are the "solutions" of the equation.

Laws of elementary algebraMirsky, Lawrence (1990) An Introduction to Linear Algebra' Library of Congress''. p.72-3. ISBN 0-486-66434-1.

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• Addition is a commutative operation (two numbers add to the same thing whichever order you add them in).
  • Subtraction is the reverse of addition.
  • To subtract is the same as to add a negative number:

  $a\; -\; b\; =\; a\; +\; (-b).\; \backslash$

  Example: if $5\; +\; x\; =\; 3$ then $x\; =\; -2.$

  • Multiplication is a commutative operation.

• Division is the reverse of multiplication.
• To divide is the same as to multiply by a reciprocal:

$\{a\; \backslash over\; b\}\; =\; a\; \backslash left(\; \{1\; \backslash over\; b\}\; \backslash right).$

• Exponentiation is not a commutative operation.

• Therefore exponentiation has a pair of reverse operations: logarithm and exponentiation with fractional exponents (e.g. square roots).
  • Examples: if $3^x\; =\; 10$ then $x\; =\; \backslash log\_3\; 10\; .$ If $x^\{2\}\; =\; 10$ then $x\; =\; 10^\{1\; /\; 2\}.$
• The square roots of negative numbers do not exist in the real number system. (See: complex number system)

Associative property of addition: $(a\; +\; b)\; +\; c\; =\; a\; +\; (b\; +\; c).$

Associative property of multiplication: $(ab)c\; =\; a(bc).$

Distributive property of multiplication with respect to addition: $c(a\; +\; b)\; =\; ca\; +\; cb.$

Distributive property of exponentiation with respect to multiplication: $(a\; b)^c\; =\; a^c\; b^c\; .$

How to combine exponents: $a^b\; a^c\; =\; a^\{b+c\}\; .$

Power to a power property of exponents: $(a^b)^c\; =\; a^\{bc\}\; .$

Laws of equality

• If $a\; =\; b$ and $b\; =\; c$, then $a\; =\; c$ (transitivity of equality).
• $a\; =\; a$ (reflexivity of equality).
• If $a\; =\; b$ then $b\; =\; a$ (symmetry of equality).

Other laws

• If $a\; =\; b$ and $c\; =\; d$ then $a\; +\; c\; =\; b\; +\; d.$
  • If $a\; =\; b$ then $a\; +\; c\; =\; b\; +\; c$ for any c (addition property of equality).
• If $a\; =\; b$ and $c\; =\; d$ then $ac$ = $bd.$
  • If $a\; =\; b$ then $ac\; =\; bc$ for any c (multiplication property of equality).
• If two symbols are equal, then one can be substituted for the other at will (substitution principle).
• If $a\; >\; b$ and $b\; >\; c$ then $a\; >\; c$ (transitivity of inequality).
• If $a\; >\; b$ then $a\; +\; c\; >\; b\; +\; c$ for any c.
• If $a\; >\; b$ and $c\; >\; 0$ then $ac\; >\; bc.$
• If $a\; >\; b$ and then

Examples

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Linear equations in one variable

The simplest equations to solve have only one variable. They contain only constant numbers and a single variable without an exponent. For example:

$2x\; +\; 4\; =\; 12.\; \backslash ,$

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 unknown variable. For example, if we subtract 4 from both sides in the equation above we get:

$2x\; +\; 4\; -\; 4\; =\; 12\; -\; 4\; \backslash ,$

which simplifies to:

$2x\; =\; 8.\; \backslash ,$

Dividing both sides by 2:

$\backslash frac\{2x\}\{2\}\; =\; \backslash frac\{8\}\{2\}\; \backslash ,$

simplifies to our solution:

$x\; =\; 4.\; \backslash ,$

Quadratic equations

Quadratic equations can be expressed in the form $ax^\{2\}\; +\; bx\; +\; c\; =\; 0,$ where a cannot equal 0. (Notice if a does equal 0 we no longer have a quadratic equation but a linear one.) Because of this a quadratic equation must contain the term $ax^\{2\},$ which is known as the quadratic term. Quadratic equations be solved using factorization or the quadratic formula. As an example of factoring:

$x^\{2\}\; +\; 3x\; =\; 0.\; \backslash ,$

This is the same thing as

$x(x\; +\; 3)\; =\; 0.\; \backslash ,$

Setting x to 0 or -3 will make this true. (Since both $0*(x\; +\; 3)$ and $x\; *\; (-3\; +\; 3)$ equals 0). All quadratic equations will have two solutions in the complex number system, but need not have any in the real number system. For example,

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

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

$(x\; +\; 1)^\{2\}\; =\; 0.\; \backslash ,$

For this equation, -1 is a root of multiplicity 2.

System of linear equations

If we have a system of linear equations, for example, two equations in two variables, it is often possible to find two answers that satisfy both.

$4x\; +\; 2y\; =\; 14\; \backslash ,$

$2x\; -\; y\; =\; 1.\; \backslash ,$

Now, multiply the second equation by 2 on both sides, and you have the following equations:

$4x\; +\; 2y\; =\; 14\; \backslash ,$

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

Now we add the two equations together to get:

$8x\; =\; 16\; \backslash ,$

$x\; =\; 2.\; \backslash ,$

You can see that since we multiplied the second equation by 2, we can combine the equations and cancel out y, and then we can solve for x. Note that you can multiply by any numbers (positive or negative, but not zero) to both sides of any to get to a point where a variable cancels out when you combine them.

To find y, choose either one of the equations from the beginning.

$4x\; +\; 2y\; =\; 14.\; \backslash ,$

Substitute in 2 for x.

$4(2)\; +\; 2y\; =\; 14.\; \backslash ,$

Simplify using the rules of algebra.

$8\; +\; 2y\; =\; 14\; \backslash ,$

$2y\; =\; 6\; \backslash ,$

$y\; =\; 3.\; \backslash ,$

The full solution to this problem is then

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

See also

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• Binomial
• Polynomial
• Vulgar fraction
• Number line
• FOIL rule
• Mathomatic

References

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• Charles Smith, A Treatise on Algebra, in Cornell University Library Historical Math Monographs.
• Beginning Algebra Tutorials and Reviews for basic algebra review and practice..
• Feferman, Anita Burdman and Solomon Feferman (1990) "Alfred Tarski- Life and Logic." Cambridge University Press. p.74-76. ISBN 0-521-80240-7.

Further reading

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• Other example problems can be found at www.exampleproblems.com.

Footnotes

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Elementary algebra | School subjects

جبر ابتدائي | Elementare Algebra | Algèbre élémentaire | Algebra elementare | Elementaire algebra | Álgebra elementar | Elementär algebra | அடிப்படை இயற்கணிதம் | Елементарна математика