31.10 - Variables

Definition

Variable: (derived from Latin variabilis, meaning ‘changeable’) is a symbol, typically a letter that refers to an unspecified mathematical object, like a numerical value. May also be called a parameter in some cases

Take this example:

Example

You’re working for 4 euros per hour. The number of hours you work is variable. If you work 3 hours on a day, you earn $4\times 3=12$ euro. If you work 5 hours on a day, you earn $4\times 5=20$ euro.

In this scenario, you could substitute the $4$ euros per hour for $x$(or any letter for that matter, though some are used less often than others).
In the example above, $4\times x$ is called an algebraic expression and $x$ is called a variable.

Sums, products and variables

Sums

The expression $x+3y$ is the sum of $x$ and $3y$. In this example, we call $x$ and $3y$ the terms of the sum.

Differences

In case there is a minus in the expression, the minus will be added to the number/variable on the right when writing down the terms.

Example

Take the following expression: $4c^2-3d$ The terms are $4c^2$ and $(-3)d$

Products

Whenever you multiply a variable, instead of using the standard multiplication sign, you use the multiplication dot, denoted as $\cdot$ (\cdot in LaTeX).

The dots are optional, meaning you don’t have to write them.
All the symbols(numbers and letters) in the product are called the factors.
All the numbers in the product can also be called the coefficients. Any variable without a coefficient would be the same as putting a 1 in front of it: $x=1x$

Example

Instead of writing: $3\times x\times y$, we write: $3\cdot x\cdot y$. Or even: $3xy$ $3$, $x$ and $y$ are called the products $3$ is called the coefficient

Substitution

Substitution is the process of switching out variables in an expression for values.

Example

Calculate $x^2+x$ for $x=3$: $x^2+x=3^2+3=12$

Simplifying expressions with variables

Expressions aren’t always written in the shortest/clearest way. Sometimes products can be re-arranged or sums can be added together.
This process is called simplifying.

Simplifying usually consists of multiplying coefficients in the same product, or adding coefficients of different terms of a sum/difference, who have the same variable.

Example

Take the following expression: $5\cdot x+2\cdot x\cdot 8$ First we simplify the multiplication: $5\cdot x+16\cdot x$, because $2\times 8=16$ Then, we simplify the addition: $21x$, because $5+16=21$

Simplifying multiplication

Usually when expressions have multiple coefficients within the same term, you can multiply them together, and since every number/variable in a product is re-arrangeable, you can make expressions look more readable.

Example

$4\cdot 5\cdot x\cdot 3\cdot y$ $4\times 5\times 3=60$ thus, $4\cdot 5\cdot x\cdot 3\cdot y=60xy$

Simplifying addition/subtraction

Whenever the two terms of a sum/difference have identical variables, you can add their coefficient together. Simply remove the variables, solve the sum/subtraction, and add back the variable.

Beware of factors

Keep in mind that variables with factors are different from variables without them. $x$ is not the same term as $x^2$

Example

Take the following sum: $6x+2x-3x$. This can be simplified into: $6x+2x-3x$ > $6+2-3=5$ > $5x$

Algebraic rules

Algebraic rules are a set of rules that makes simplifying expressions a bit faster. Some of them are already explained in the previous section.

Here’s some of them:

rules

• All terms of a sum/difference can be re-arranged
• the same is true for a product
• $x+0=x$
• $1\cdot x=x$
• $-1\cdot x=-x$
• $0\cdot x=0$