Algebra

Algebra is the intellectual instrument which has been created for rendering clear the quantitative aspects of the world. —Alfred North Whitehead

Algebra is a part of mathematics. Variables can represent unknown or changing values. An equation states that two expressions, separated by an equal sign (=), have the same value. A very simple equation using a variable is: 2 + 3 = x, in this example, x = 5 (or it could also be said, x equals five). This is called solving for x.

Inequalities compare expressions using relations such as less than or greater than. A function assigns exactly one output to each input in its domain. Equations can describe functions, and graphs can display their input-output relationships.

Algebra can be used to solve real problems because the rules of algebra work in real life, and numbers can be used to represent the values of real things. Physics, engineering and computer programming are areas that use algebra all the time. It is also useful in surveying, construction and business—especially accounting.

People who apply algebra use the rules of numbers and mathematical operations. The simplest ones are adding, subtracting, multiplying, and dividing. More advanced operations involve exponents, starting with squares and square roots.

Algebra was first used to solve equations and inequalities. Two examples are linear equations (the equation of a straight line, y = mx + b) and quadratic equations, whose highest variable power is the square (a quantity multiplied by itself, for example: 2 * 2, 3 * 3, or x * x).

Example equation solving: 4x - 8x + 2 = -7 - 6x + 12 → 4x - 8x + 6·x = -7 + 12 - 2 → 2x = 3 → x = 3/2

Simple Word Problem

Sue has 12 candies, and Ann has 24 candies. They decide to combine and share these so that they have the same number of candies each. These are the steps you can use to solve the problem:

1. To have the same number of candies, Ann has to give some to Sue. Let x represent the number of candies Ann gives to Sue.
2. Sue’s candies, plus x, must be the same as Ann’s candies minus x. This is written as: 12 + x = 24 - x
3. Subtract 12 from both sides of the equation. This gives: x = 12 - x. (What happens on one side of the equals sign must happen on the other side too, for the equation to still be true. So, in this case, when 12 was subtracted from both sides, there was a middle step of 12 + x - 12 = 24 - x - 12. After a person is comfortable with this, the middle step is not written down.)
4. Add x to both sides of the equation. This gives: 2x = 12
5. Divide both sides of the equation by 2. This gives x = 6. The answer is six. If Ann gives Sue 6 candies, they will have the same number of candies.
6. To check this, put 6 back into the original equation wherever x was: 12 + 6 = 24 – 6
7. This gives 18 = 18, which is true. They both now have 18 candies.

Algebra helps express constraints and relationships in problems such as planning a structure, designing electronics, or analyzing experimental measurements. Solving the equation is useful only if the model represents the relevant features of the problem.

Rules of Algebra

Two useful definitions:

• Opposite: the opposite of a is -a
• Reciprocal: the reciprocal of a is 1/a, provided a is nonzero

Commutative Property of Addition

An operation is commutative if swapping its two inputs leaves the result unchanged. For addition: a + b = b + a

Commutative Property of Multiplication

When the operator of two terms is a multiplication, the commutative property of multiplication is applicable. In algebraic terms, this gives a · b = b · a

Associative Property of Addition

Associative refers to the grouping of numbers. The associative property of addition implies that, when adding three or more terms, it does not matter how these terms are grouped. Algebraically, this gives: a + (b + c) = (a + b) + c.

Associative Property of Multiplication

The associative property of multiplication implies that, when multiplying three or more terms, it does not matter how these terms are grouped. Algebraically, this gives: a · (b · c) = (a · b) · c

Distributive Property

The distributive property states that the multiplication of a number by another term can be distributed: a · (b + c) = a · b + a · c

Additive Identity Property

The additive identity leaves a number unchanged when added to it: the sum of any number and 0 is that number: a + 0 = a. This also holds for subtraction a - 0 = a

Multiplicative Identity Property

The multiplicative identity property states that the product of any number and 1 is that number a · 1 = a. This also holds for division a / 1 = a

Additive Inverse Property

A number plus its additive inverse equals 0. Equivalently: a – a = 0. Additive inverse of 1 is (-1)

Multiplicative Inverse Property

A nonzero number multiplied by its reciprocal equals 1. Equivalently: a / a = 1. Multiplicative inverse of 2 is 1/2