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# Elementary algebra

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 + 2 = 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; examples are a + 3 and x2 - 3. 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: x2 - 1 = 4. These are the "solutions" of the equation.

As in arithmetic, in algebra it is important to know precisely how mathematical expressions are to be interpreted. This is governed by the order of operations rules.

It is then necessary to be able to simplify algebraic expressions. For example, the expression

$-4(2a + 3) - a \,$

can be written in the equivalent form

$-9a - 12 \,$.

The simplest equations to solve are the linear ones, such as

$2x + 3 = 10 \,$

The central technique is add/subtract/multiply or divide both sides of the equation by the same number, and by repeating this process eventually arrive at the value of the unknown x. For the above example, if we subtract 3 from both sides, we obtain

$2x = 7 \,$

and if we then divide both sides by 2, we get our solution

$x = \frac{7}{2}$

Equations like

$x^{2} + 3x = 5 \,$

are known as quadratic equations and can be solved using the quadratic formula.

Expressions or statement may contain many variables, from which you may or may not be able to deduce the values for some of the variables. For example:

$(x - 1)^{2} = 0y \,$

After some algebraic steps (not covered here), we can deduce that x = 1, however we cannot deduce what the value of y is. Try some values of x and y (which may lead to either true or false statements) to get a feel for this.

However, if we had another equation where the values for x and y were the same, we could deduce the answer in a process known as systems of equations. For example (assume x and y are the same values in both equations):

$4x + 2y = 14 \,$
$2x - y = 1 \,$

Now, multiply the second by 2, and you have the following equations:

$4x + 2y = 14 \,$
$4x - 2y = 2 \,$

Because we multiplied the entire equation by 2, it actually represents the same statement. Now we can combine the two equations:

$8x = 16 \,$

You can see that since we multiplied the second equation by 2, we can cancel out y when combining the equations, and then we can solve for x, which is 2. Note that you can multiply by negative numbers, or multiply both equations to get to a point where a variable cancels out (you can also cancel out x).

Now choose one of the equations from the beginning.

$4x + 2y = 14 \,$

Substitute in 2 for x.

$4(2) + 2y = 14 \,$

Simplify.

$8 + 2y = 14 \,$
$2y = 6 \,$

And solve for y, which equals 3. The answer to this problem is x = 2 and y = 3, or (2,3).

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

## Laws of elementary algebra

$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 reciprocal exponents (e.g. square roots).
• Examples: if 3x = 10 then x = log310. If x2 = 10 then x = 101 / 2.
• The square root of negative one is i.
• Distributive property of multiplication with respect to addition: c(a + b) = ca + cb.
• Distributive property of exponentiation with respect to multiplication: (ab)c = acbc.
• How to combine exponents: abac = ab + c.
• 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).
• If a = b and c = d then a + c = b + d.
• If a = b then a + c = b + c for any c, due to Reflexivity of Equality.
• If a = b and c = d then ac = bd.
• If a = b then 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 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.