# 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 $\displaystyle a + b = b + a$ 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 $\displaystyle 3x + 2 = 10$ ).
• It allows the formulation of functional relationships (such as "if you sell x tickets, then your profit will be $\displaystyle 3x - 10$ 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:

$\displaystyle x + 3\,$
$\displaystyle y^{2} - 3\,$
$\displaystyle z^{7} + a(b + x^{3}) + 42/y - \pi.\,$

An "equation" is the claim that two expressions are equal. Some equations are true for all values of the involved variables (such as $\displaystyle a + (b + c) = (a + b) + c$ ); 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: $\displaystyle x^{2} - 1 = 4.$ These are the "solutions" of the equation.

## Laws of elementary algebra

$\displaystyle a - b = a + (-b). \$
Example: if $\displaystyle 5 + x = 3$ then $\displaystyle x = -2.$
$\displaystyle {a \over b} = a \left( {1 \over b} \right).$
• 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 $\displaystyle 3^x = 10$ then $\displaystyle x = \log_3 10 .$ If $\displaystyle x^{2} = 10$ then $\displaystyle x = 10^{1 / 2}.$
• The square roots of negative numbers do not exist in the real number system. (See: complex number system)
• Associative property of addition: $\displaystyle (a + b) + c = a + (b + c).$
• Associative property of multiplication: $\displaystyle (ab)c = a(bc).$
• Distributive property of multiplication with respect to addition: $\displaystyle c(a + b) = ca + cb.$
• Distributive property of exponentiation with respect to multiplication: $\displaystyle (a b)^c = a^c b^c .$
• How to combine exponents: $\displaystyle a^b a^c = a^{b+c} .$
• If $\displaystyle a = b$ and $\displaystyle b = c$ , then $\displaystyle a = c$ (transitivity of equality).
• $\displaystyle a = a$ (reflexivity of equality).
• If $\displaystyle a = b$ then $\displaystyle b = a$ (symmetry of equality).
• If $\displaystyle a = b$ and $\displaystyle c = d$ then $\displaystyle a + c = b + d.$
• If $\displaystyle a = b$ then $\displaystyle a + c = b + c$ for any c, due to reflexivity of equality.
• If $\displaystyle a = b$ and $\displaystyle c = d$ then $\displaystyle ac$ = $\displaystyle bd.$
• If $\displaystyle a = b$ then $\displaystyle ac = bc$ for any c due to Reflexivity of Equality.
• If two symbols are equal, then one can be substituted for the other at will.
• If $\displaystyle a > b$ and $\displaystyle b > c$ then $\displaystyle a > c$ (transitivity of inequality).
• If $\displaystyle a > b$ then $\displaystyle a + c > b + c$ for any c.
• If $\displaystyle a > b$ and $\displaystyle c > 0$ then $\displaystyle ac > bc.$
• If $\displaystyle a > b$ and $\displaystyle c < 0$ then $\displaystyle ac < bc.$

## 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:

$\displaystyle 2x + 4 = 12. \,$

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:

$\displaystyle 2x = 8 \,$

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

$\displaystyle x = \frac{8}{2} = 4.$

Quadratic equations contain variables raised to the first and second (square) power, and can be solved using factorization or the quadratic formula. As an example of factoring:

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

This is the same thing as

$\displaystyle x(x + 3) = 0. \,$

Setting x to 0 or -3 will make this true. All quadratic equations will either have one or two solutions.

### 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.

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

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

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

Now we add the two equations together to get:

$\displaystyle 8x = 16 \,$
$\displaystyle x = 2. \,$

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 when you combine them.

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

$\displaystyle 4x + 2y = 14. \,$

Substitute in 2 for x.

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

Simplify using the rules of algebra.

$\displaystyle 8 + 2y = 14 \,$
$\displaystyle 2y = 6 \,$
$\displaystyle y = 3. \,$

The full solution to this problem is then

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