Expressions And Equations

Expressions and Equations are fundamental concepts in algebra. They help us represent mathematical relationships and solve problems in both mathematics and real-life situations. Understanding these concepts builds the foundation for solving simple as well as complex algebraic problems.

Expression Definition

An algebraic expression is a mathematical combination of numbers, variables (letters that represent numbers), and arithmetic operations such as addition, subtraction, multiplication, and division. It represents a value and does not contain an equality sign (=).

Types of Expressions

There are different types of expressions. Some of them are mentioned below:

Numerical Expressions

These expressions contain only numbers and operations.

Example: 7 + 8 × 3

Algebraic Expressions

These expressions contain variables, numbers, and operations

Example: 3x² + 7x - 3

Polynomial Expressions

These expressions contain algebraic expressions with multiple terms.

Example: 4x³ - 3x² + 2x - 4

Rational Expressions

Expressions which have division of two or more polynomials are called as rational expressions.

Example: 1/x + 2/x + 3

Radical Expressions

Expressions which Include variables or numbers under a root sign are called as radical expressions.

Example: √3x + 7

Equation Definition

An equation is a mathematical statement that maintain the equality of two expressions, separated by an equality sign =. We solve these equation to find the value of variable that make the equation true.

Example: x² - 3x -3 = 0

This is an example of equation where x is variable and 1 , -3 and -3 are constants.

Types of Equation

Linear Equations

Equations of the first degree i.e. the highest exponent of the variable is 1 is known as linear equation.

Example: 3x + 4 = 12

Quadratic Equations

Equations of the second degree i.e. the highest exponent of the variable is 2 is known as quadratic equation.

Example: 3x² - 4x + 12 = 0

Polynomial Equations

Equations which have polynomial expressions is known as polynomial equation.

Example: 4x⁴ + 3x² + 1 = 0

Rational Equations

Equations which involve rational expression such as fraction.

Example: 1/x + 2/x+1 = 3

Expression vs Equation

Below is the key differences between expressions and equations in tabular form:

Solved Examples

Example 1: Simplify the given expression: 2x+3x-4+7.

Solution:

We combine like terms:

2x + 3x = 5x -4 + 7 = 3

So, simplified expression is 5x + 3.

Example 2: Solve the following equation: 3x - 5 = 7

Solution:

3x - 5 = 7

Add 5 to both side of equation

3x - 5 + 5 = 7 + 5

3x = 12

x = 12/3

x = 4

Example 3: Factor the quadratic equation x² - 5x + 6 = 0.

Solution:

Find two numbers that multiply to 6 and add to -5: these two numbers are -2 and -3 (x - 2) ( x - 3) = 0

So, factors are (x - 2) ( x - 3)

Example 4: Factor the quadratic equation x² - 4x + 4 = 0.

Solution:

We can identify a pattern in this question:

(x)² - (2)(2)(x) + (2)²

It is a question of pattern (a - b)² = a² -2ab + b

(x - 2)² = 0

So, factors are (x - 2) ( x - 2)

Example 5: Solve the linear Equation 5x - 7 = 3x + 9.

Solution:

To solve the given linear term of given equation:

5x - 7 = 3x + 9

Subtract 3x from both sides

5x - 3x - 7 = 9

2x - 7 = 9

Add 7 to both sides:

2x - 7 + 7 = 9 + 7

2x = 16

Divide by 2:

x = 16/2 = 8

Related Articles

• Expressions in Math
• Equation in Maths
• Algebraic Expressions in Math: Definition, Example and Equation
• Equations with Variables