Expression Math

Defining an Expression

An expression can be used to describe a scenario when an unknown Number is present or when a variable value exists. It helps solve real-world problems in a more simplified and explicit manner.

To construct an expression of this kind, you would need to determine which quantity is unknown in the circumstance, and then define a variable to represent it. Before we dive into this topic further, let us first define expressions.

Let's see an example of an expression.

Expressions are very organised, in a way that a statement that has an operator come right after another one is not a valid expression. For example,

\[2x+\times 1.\]

They are also organised in the sense that when a parenthesis opens, there needs to be a close. For example,

\[3(4x+2)-6\]

is a valid expression. However,

\[6-4(18x\]

is not a valid expression.

Components of an Expression

Expressions in Algebra contain at least a variable, numbers, and an arithmetic operation. However, there are quite a Number of terms related to the parts of an expression. These elements are described below.

• Variables: Variables are the letters that represent an unknown value in a mathematical statement.

• Terms: Terms are either numbers or variables (or numbers and variables) multiplying and dividing each other and are separated by either the addition (+) or subtraction sign (-).

• Coefficient: Coefficients are the numbers that multiply variables.

• Constant: Constants are the numbers in expressions that do not change.

Components of an expression

Examples of Expressions

Here are some examples of mathematical expressions.

Notice that all of them contain the necessary components to be considered expressions. They all have variables, numbers, and at least one mathematical operation composing them.

In particular, in the first example, you will find a multiplication implicit in the parenthesis that connects the two terms \(x+1\) and \(x+3\); so it is a valid expression. In the fourth example, in the second term, variables \(x\) and \(y\) are multiplying and it's written as \(xy\). So, that one is also a valid expression.

Writing Expressions

In this segment of our discussion, we will be introduced to writing expressions, particularly translating word problems into mathematical ones. Such skill is important when solving a given question. By doing so, we can visualise anything in terms of numbers and arithmetic operations!

Translating Word Problems into Expressions

Given a sentence that illustrates a mathematical statement, we can translate them into expressions that involve the appropriate components of expressions we had mentioned before and mathematical symbols. The table below demonstrates several examples of word problems that have been translated into expressions.

Types of Maths Expressions

Numerical Expressions

In comparison to what expressions are, there are expressions that do not contain variables. These are called numerical expressions.

They could be as long as possible, containing as many mathematical operators as possible also.

Here are a few examples of numerical expressions.

Algebraic Expressions

Algebraic expressions are expressions that contain unknowns. Unknowns are variables that are often represented by letters. In most cases throughout our syllabus, these letters are \(x\), \(y\) and \(z\).

However, we may sometimes get expressions that comprise Greek letters as well. For instance, \(\alpha\), \(\beta\) and \(\gamma\). Below are several examples of algebraic expressions.

Evaluating Maths Expressions

In this section, we will be introduced to evaluating math expression. Here, we would essentially solve a given expression based on the arithmetic operations between the numbers or variables. These basic arithmetic operations (or mathematical symbols) include Addition, Subtraction, Multiplication and Division. We will also see how these operations can help us factorise and simplify such expressions.

Addition and Subtraction of Expressions

Addition and subtraction are the primary actions done when adding and subtracting Fractions. These are performed on like terms. There are two steps to consider here, namely

• Step 1: Identify and rearrange like terms to be grouped.

• Step 2: Add and subtract like terms.

Below is a worked example.

Here is another worked example for you.

Factorising Expressions

This is an important element when it comes to dealing with expressions. It helps us group like terms in order for us to perform arithmetic operations more structured manner.

The factorised form of expressions is always in brackets. The process involves taking out the highest Common Factors (HCF) from all the terms such that when the Factors are taken out and multiplied by the values in the brackets, we will arrive at the same expression we had in the first place.

For example, say you had the expression below.

\[4x^2+6x\]

Notice here that the coefficients of \(x^2\) and \(x\) both have a factor of 2 since 4 and 6 are divisible by 2. Furthermore, \(x^2\) and \(x\) have a common factor of \(x\). Thus, you can take these two Factors out of this expression, making the factories form equivalent to

\[2x(2x+3)\]

Let's explain this again with another example.

Let's go through one more example.

Simplifying Expressions

The term simplifying stems from the root word "simple". As the word suggests, simplifying a given expression allows us to solve them more efficiently. When we simplify an expression, we are reducing it into a simpler form by cancelling Common Factors and regrouping terms that share the same variable.

This avoids all the lengthy working you might have to perform that may result in unwanted careless mistakes. Surely, you wouldn't want to have any arithmetic errors now, would you?

There are three steps to follow when simplifying expressions.

1. Eliminate the brackets by multiplying out the factors (if any are present);

2. Remove exponents by using the exponent rules;

3. Add and subtract like terms.

Let's go through some worked examples.

Here is another example.