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# Expression|Definition & Meaning

## Definition

An expression is a **mathematical** sentence consisting of numbers, **variables**, and math **operators.** It should have at least one math operator and two **terms**, numbers, or variables to be called an expression.

A math **operation** is performed in an **expression** between different terms, resulting in another **term** or value. **Figure 1** shows a primary expression.

The **addition** operation is performed between two **unlike** terms 7a and b.

## Components of an Expression

The following are required to make an **expression** in mathematics.

### Numbers

Numbers are an important part of expressions. A math **operation** performed on **two** or more numbers is a basic **expression.** If a number is multiplied by a variable, it is known as the **coefficient**.

### Variables

A variable is an **unknown** quantity in an expression. An **alphabet** or any **symbol** denotes it. Expression with variables is used in **Algebra** either to **simplify** them or to solve them for the **unknown** variables in equation form.

### Terms

A term in mathematics is defined as a **number,** a variable, a number multiplied by a **variable,** many variables multiplied by each other, or variables **multiplied** by a number.

A variable should only be multiplied and not **added** or **subtracted** to form a term. For example, **5**,** x**, **2z**, **abc**, and **4lmn** are different types of **terms.**

An **expression** can consist of a single term or **multiple** terms. There are two types of terms.

#### Like Terms

Like terms are the **terms** having the same **variable** with the same **power.** For example, **8b** and **2b** are like terms having the variable **b** with power** 1**.

These terms can be **simplified** in an expression by adding or subtracting.

#### Unlike Terms

The terms having **different** variables or the same **variables** with different **powers** are known as, unlike terms. For example, **4x** and **2y** are unlike terms. Similarly, 3c$\mathsf{^2}$ and 5c$\mathsf{^3}$ are also **unlike** terms.

An **expression** consisting of unlike terms cannot be simplified.

### Math Operators

Math operators have a unique **function** applied to elements to produce a certain **result**. The basic operators are **addition,** subtraction, multiplication, and **division.**

Multiplication is a form of repeated **additions** and division is a form of repeated **subtraction.**

An **expression** cannot be formed without an **operator**. The terms on which the operation is applied are also known as the **operands.**

**Figure 2** shows all the **components** of an **expression**.

## Types of Expressions

There are **three** types of expressions.

### Numeric Expressions

An expression consisting of only **numbers** is known as a **numeric** expression. The math operation is done on numbers also known as **constants**.

For example, **9(2 + 5)** is a numeric expression with **addition** and **multiplication** as its operators.

### Fractional Expressions

These expressions have **fractions** as their terms. For example, **2/3 – 1/4** is a fractional expression.

### Algebraic Expressions

Expressions consisting of **variables**, operators, and **constants** are known as **algebraic** expressions. For example, **7x + 6 – 5y** is an algebraic expression with variables** x** and **y**.

**Figure 3** shows the types of **expressions**.

## Types of Algebraic Expressions

Algebraic expressions have an essential role in **Algebra**. Depending on the **number** of **terms**, there are **four** main types of algebraic expressions.

### Monomial Expression

A **monomial** expression consists of only **one** algebraic term. For example, **3x** is a monomial expression.

### Binomial Expression

“**Bi**” means **two**. So, a binomial expression consists of two **terms** only. For example, **3 + 2z** is a one-variable **binomial** expression of variable **z**.

### Trinomial Expression

A trinomial **expression** consists of **three** terms. For example, **p + 3q + 6r** is a trinomial expression of three variables **p**, **q**, and **r**.

### Polynomial Expression

An **expression** consisting of **more** than three terms is known as a **polynomial** expression. The expression **2l + 5m + n – 7r + 6** is a polynomial with variables** l**, **m**, **n**, and** r**.

**Figure 4** shows the four types of **algebraic** expressions.

## Expressions in Equations

An **equation** is formed with expressions. It consists of an **equality** sign “**=**” and can have two **expressions** on both of its sides. **Figure 5** shows an **equation** with an expression on its left-hand side.

The above **equation** can be solved by adding **9** on both sides which makes the right-hand side an **expression** too.

2x – 9 + 9 = 11 + 9

2x = 20

Dividing **2** into both sides gives the value of** x:**

**x = 10**

Hence, **expressions** are used in equations to solve for unknown **variables**.

## Rules To Solve Expressions

The following **acronyms** are used to solve expressions.

### PEMDAS

**PEMDAS** stands for “Parenthesis Exponent Multiplication Division Addition Subtraction”. This is the **order** of **operations** followed to solve the mathematical expressions.

The highest priority is of solving **parentheses**, then the terms with **powers**, and so on.

### BIDMAS

Another **acronym** used is BIDMAS. It **stands** for “Brackets Index Division Multiplication Addition Subtraction”. It is the same as **PEMDAS**.

It is used for **expressions** consisting of more than one math **operations**.

## Extracting Expressions From Sentences

Word **problems** contain sentences that need to be **translated** into mathematical **expressions** to solve. For example, the sum of** 3** and **8** forms an expression as **3 + 8**.

If **2** is **subtracted** from **4x**, the expression will be **4x – 2**. The product of** 3** and **6x** is expressed as **3(6x). 3** divided by **4** will be written as **3/4**.

## An Example of Evaluating Expressions

Solve the following **algebraic** expressions:

a) 5x + 8 – 2y + 3x – 6y – 2

b) 2(3l + 5m) – 3(l + 4m)

### Solution

#### a) 5x + 8 – 2y + 3x – 6y – 2

At first, the **like terms** should be placed together as:

5x + 3x – 2y – 6y + 8 – 2

Solving the **like** terms gives:

8x – 8y + 6

The **final** expression is **8x – 8y + 6**.

#### b) 2(3l + 5m) – 3(l + 4m)

By the **distributive** law, solving the **parentheses** gives:

6l + 10m – 3l – 12m

Solving the **like** terms gives:

6l – 3l + 10m – 12m

3l – 2m

So, the **simplified** expression is** 3l – 2m**.

*All the images are created using GeoGebra.*