Intermediate Algebra/Variables

Template:Algebra Page The grammar of algebra describes how symbols are grouped together to express meaning in algebraic expressions. It is much easier to learn algebra when you know what the symbols mean.

Variables and Constants

We will deal with symbols that represent unspecified numbers. They serve as placeholders for numbers that we do not know, or numbers that vary. These symbols are called variables. Symbols that represent unchanging and specific numbers are called constants.

Variables are typically written using letters, such as x, t, or A. Constants are typically written as the numbers themselves, such as 2, -5, and 0.75, or as special reserved characters, like π.

(Exercise) Decide whether the following are variables or constants: 1, 2, 3, a

Answer: Since the symbols 1, 2, and 3 represent specific numbers, they are constants. Since a represents an unspecified number, it is a variable.

Dependent and Independent Variables

Dependent variables depend on other variables. Their value changes based on the value of something else. Independent variables vary, but they do not change based on the value of other variables.

(Exercise). There are a number of litres of fuel in a gas tank, represented by f. The speedometer points at a number, represented by s. The fuel gauge is pointing at a number, represented by g. Of f, s, and g, which is dependent, and which is independent?

Answer: The gauge's reading (g) depends on the amount of fuel in the tank (f). g is dependent, while f is independent. s, the speedometer, is irrelevant.

By convention, we call the dependent variable y and place it alone and on the left side of the equation. We call the independent variable x and place it on the right.

Y = x + 2

Here, both Y and x are variables, while 2 is a constant. Without more information (what does Y, or x, represent?), we can't know for sure which variable is which, but because of the convention, we can safely assume that Y is dependent, while x is independent.

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• A coefficient is a variable or number that is multiplied by something other than itself

In the equation y = 3x + 7, the 3 is the coefficient of x, and the 7 is a constant.

Terms

In algebra a "term", or monomial, is made up of any number of constants, variables, exponents, etc. that are multiplied with one another. More technically, a monomial is made up of any member of a family of functions multiplied by any number of members of other families of functions. (More on functions and families later.)

Some examples of monomials:
7
7x
7x²
7x²y³

Notice the identifying feature of the monomial is that there is no addition or subtraction. Therefore, 7x + 2y would not be a monomial.

Like Terms, generally, are monomials that have two things in common:
1) They are composed of the same variables.
2) Those variables share the same exponents.
Note: Constants do not matter when identifying like terms.

If two monomials are "like", they can be added/subtracted. This is often referred to combining like terms.
Example: 7x + 2xy + 3y² + 4x + 4y³

The only terms that are "like" are 7x and 4x. This means we can add them together into one term, 11x. 3y² and 4y³ are close, but since the y parts have different exponents, they are not like terms.

When adding like terms, only change the constants. The exponents will not change.
Example: 3x² + 2x² + 4x²y² + 5x²y²
Combining like terms will give us 5x² + 9x²y²

Combining like terms is important in mathematics because it gives us the simplest, least cluttered solution. Here is one last example:
3x + 7x²y⁴ + 5x⁴y² + 2x³ + 8y⁵ + 3xyz
None of the monomials are like terms, so we are as simple as we can get.

Algebraic Expressions

Algebraic expressions are a combination of terms connected by operators. Examples include $\frac{2}{3}$, $x-2$, $\frac{a}{b}+8$, and $\frac{y-b}{m}$.

The Distributive Property

Multiplying a given number by a sum yields the same result as multiplying the number by each addend in the sum, then adding the resultant products.*

This may be hard to grasp when reading about it in the abstract, but the following example should help:

3(6+2) = 3(8) = 24

3(6+2) = 3(6) + 3(2) = 18 + 6 = 24

We will be using the Distributive Property a lot when solving algebraic problems. It can be used to "switch" between equivalent representations of unknown quantities, which gives us a great deal of flexibility, since we can pick the representation that is easiest to work with when dealing with a particular problem. Take a look at the example below:

3(x+2) = 3x+3(2) = 3x+6

7a + 2a = (7 + 2)a = 9a (this is the rationale for adding "like" (or "similar") terms)

Depending on the situation, either the expression on the right or the left may be more useful.