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Elementary algebra

2007 Schools Wikipedia Selection. Related subjects: Mathematics

   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. While in arithmetic only
   numbers and their arithmetical operations (such as +, −, ×, ÷) occur,
   in algebra one also uses symbols (such as x and y, or a and b) to
   denote numbers. These are called variables. This is useful because:
     * It allows the generalization of arithmetical equations (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 system.
     * It allows reference to numbers which are not known. In the context
       of a problem, a variable may represent a certain value of which is
       uncertain, but may be solved 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").

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

   In elementary 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\,

          y^{2} + 2x - 3\,

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

   In more advanced algebra, an expression may also include elementary
   functions.

   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 = b + a); such equations are called " identities". Other equations
   are true for only some values of the involved variables: x^2 − 1 = 4.
   The values of the variables which make the equation true are called the
   "solutions" of the equation.

Laws of elementary algebra

     * 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). \

                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 \over b} = a \left( {1 \over b} \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).
               o Examples: if 3^x = 10 then x = 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: (ab)^c = a^cb^c.
     * How to combine exponents: a^ba^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 c < 0 then ac < bc.

Examples

Linear equations in one variable

   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. For example:

          2x + 4 = 12. \,

   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:

          2x + 4 - 4 = 12 - 4 \,

   which simplifies to:

          2x = 8. \,

   Dividing both sides by 2:

          \frac{2x}{2} = \frac{8}{2} \,

   simplifies to the solution:

          x = 4. \,

   The general case,

          ax + b = c

   follows the same format for the solution:

          x=\frac{c-b}{a}

Quadratic equations

   Quadratic equations can be expressed in the form ax^2 + 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 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 this
   standard form.

          x^2 + px = q

   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:

          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.

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.

First method of finding a solution

   An example of a system of linear equations could be the following:

          \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.

Second method of finding a solution

   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:

          2x - 2x - y = 1 - 2x \,
          - y = 1 - 2x \,

   and multiplying by -1:

          y = 2x - 1. \,

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

          4x + 2(2x - 1) = 14 \,
          4x + 4x - 2 = 14 \,
          8x - 2 = 14 \,

   Adding 2 on each side of the equation:

          8x - 2 + 2 = 14 + 2 \,
          8x = 16 \,

   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

   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} x + y = 1 \\ 0x + 0y = 2 \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}4x + 2y = 12 \\ -2x - y = -4 \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:

          4x + 2(-2x + 4) = 12 \,
          4x - 4x + 8 = 12 \,
          8 = 12 \,

   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:

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

   Isolating y in the second equation:

          y = -2x + 6 \,

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

          4x + 2(-2x + 6) = 12 \,
          4x - 4x + 12 = 12 \,
          12 = 12 \,

   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

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

   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.

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}x + y = 2 \\ 2x + 2y = 4\end{cases}

   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}x + y = 2 \\ 4x + 4y = 1\end{cases}

   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.

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