Elementary algebra
From Academic Kids Online Encyclopedia
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 Failed to parse (unknown error): 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 Failed to parse (unknown error): 3x + 2 = 10
).
- It allows the formulation of functional relationships (such as "if you sell x tickets, then your profit will be Failed to parse (unknown error): 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; a few examples are:
- Failed to parse (unknown error): x + 3\,
- Failed to parse (unknown error): y^{2} - 3\,
- Failed to parse (unknown error): z^{7} + a(b + x^{3}) + 42/y - \pi.\,
An "equation" is the claim that two expressions are equal. Some equations are true for all values of the involved variables (such as Failed to parse (unknown error): 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: Failed to parse (unknown error): x^{2} - 1 = 4.
These are the "solutions" of the equation.
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Laws of elementary algebra
- The order of operations in a mathematical expression are as follows:
- groupings -> exponents -> multiplication -> addition
- Addition is a commutative operation.
- Subtraction is the reverse of addition.
- To subtract is the same as to add a negative number:
- Failed to parse (unknown error): a - b = a + (-b). \
- Example: if Failed to parse (unknown error): 5 + x = 3
then Failed to parse (unknown error): x = -2.
- Multiplication is a commutative operation.
- Division is the reverse of multiplication.
- To divide is the same as to multiply by a reciprocal:
- Failed to parse (unknown error): {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).
- Examples: if Failed to parse (unknown error): 3^x = 10
- Therefore exponentiation has a pair of reverse operations: logarithm and exponentiation with fractional exponents (e.g. square roots).
then Failed to parse (unknown error): x = \log_3 10 .
If Failed to parse (unknown error): x^{2} = 10
then Failed to parse (unknown error): x = 10^{1 / 2}.
- The square root of negative number does not exist in the real number system. (See: complex number system)
- Associative property of addition: Failed to parse (unknown error): (a + b) + c = a + (b + c).
- Associative property of multiplication: Failed to parse (unknown error): (ab)c = a(bc).
- Distributive property of multiplication with respect to addition: Failed to parse (unknown error): c(a + b) = ca + cb.
- Distributive property of exponentiation with respect to multiplication: Failed to parse (unknown error): (a b)^c = a^c b^c .
- How to combine exponents: Failed to parse (unknown error): a^b a^c = a^{b+c} .
- If Failed to parse (unknown error): a = b
and Failed to parse (unknown error): b = c
, then Failed to parse (unknown error): a = c
(transitivity of equality).
- Failed to parse (unknown error): a = a
(reflexivity of equality).
- If Failed to parse (unknown error): a = b
then Failed to parse (unknown error): b = a (symmetry of equality).
- If Failed to parse (unknown error): a = b
and Failed to parse (unknown error): c = d then Failed to parse (unknown error): a + c = b + d.
- If Failed to parse (unknown error): a = b
then Failed to parse (unknown error): a + c = b + c for any c, due to reflexivity of equality.
- If Failed to parse (unknown error): a = b
and Failed to parse (unknown error): c = d then Failed to parse (unknown error): ac = Failed to parse (unknown error): bd.
- If Failed to parse (unknown error): a = b
then Failed to parse (unknown error): ac = bc for any c due to Reflexivity of Equality.
- If two symbols are equal, then one can be substituted for the other at will.
- If Failed to parse (unknown error): a > b
and Failed to parse (unknown error): b > c then Failed to parse (unknown error): a > c (transitivity of inequality).
- If Failed to parse (unknown error): a > b
then Failed to parse (unknown error): a + c > b + c for any c.
- If Failed to parse (unknown error): a > b
and Failed to parse (unknown error): c > 0 then Failed to parse (unknown error): ac > bc.
- If Failed to parse (unknown error): a > b
and Failed to parse (unknown error): c < 0 then Failed to parse (unknown error): ac < bc.
Examples
Linear equations
The simplest equations to solve are linear equations. They contain only constant numbers and a single variable without an exponent. For example:
- Failed to parse (unknown error): 2x + 4 = 12. \,
The central technique is add, subtract, multiply, or divide both sides of the equation by the same thing in such a way to eventually arrive at the value of the unknown variable. If we subtract 4 from both sides in the equation above we get:
- Failed to parse (unknown error): 2x = 8 \,
and if we then divide both sides by 2, we get our solution
- Failed to parse (unknown error): x = \frac{8}{2} = 4.
Quadratic equations
Quadratic equations contain variables raised to the first and second (square) power, and can be solved using factorization or quadratic formula. As an example of factorizing:
- Failed to parse (unknown error): x^{2} + 3x = 0. \,
This is the same thing as
- Failed to parse (unknown error): x(x + 3) = 0. \,
Setting x to 0 or -3 will make this true. Most quadratic equations have two solutions.
System of linear equations
If we have system of linear equations, for example, two equations in two variables, it is often possible to find two answers that satisfy both.
- Failed to parse (unknown error): 4x + 2y = 14 \,
- Failed to parse (unknown error): 2x - y = 1. \,
Now, multiply the second equation by 2 on both sides, and you have the following equations:
- Failed to parse (unknown error): 4x + 2y = 14 \,
- Failed to parse (unknown error): 4x - 2y = 2. \,
Now we add the two equations together to get:
- Failed to parse (unknown error): 8x = 16 \,
- Failed to parse (unknown error): x = 2. \,
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 whey you combine them.
To find y, choose either one of the equations from the beginning.
- Failed to parse (unknown error): 4x + 2y = 14. \,
Substitute in 2 for x.
- Failed to parse (unknown error): 4(2) + 2y = 14. \,
Simplify using the rules of algebra.
- Failed to parse (unknown error): 8 + 2y = 14 \,
- Failed to parse (unknown error): 2y = 6 \,
- Failed to parse (unknown error): y = 3. \,
The full solution to this problem is then
- Failed to parse (unknown error): \begin{cases} x = 2 \\ y = 3. \end{cases}\,
See also
Other example problems can be found at www.exampleproblems.com.de:Elementare Algebra fr:Algèbre élémentaire it:Algebra elementare sv:Elementär algebra uk:Елементарна математика
