Expressions And Equations

Expression and Equations are two important concepts of algebra in mathematics. We need to learn about expressions and equations to solve different types of easy and complex problems in both mathematics and real-life applications. expression is a combination of numbers, variables, and operators while an equation is a mathematical statement that maintains the equality of two expressions.

In this article, we are going to learn about expressions and equations, and also different types of expressions and equations.

What is an Expression?

An expression in mathematics is a combination of numbers, letters that represent numbers, and operations such as addition, subtraction, multiplication and division. The expression represents a value by combining all these. We don't use the equality sign (=) in expressions, it is used in equations.

Example: 3x + 4y - 7

This is an expression having x and y as variables and 3, 4 and 7 as constant numbers.

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

What is an Equation?

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

Difference Between Expression and Equation

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

Components of Expressions and Equations

Expressions and equation are made of different components. Some of them are mentioned below:

Variables: Variables are symbol which are used to represents unknown values. Variables are usually denoted by letters such as x, y and z.

Constants: Constants are fixed values that do not changes. These are numbers such as 1, 2, 3...... They can be any specific variable that represent fixed known quantities.

Coefficients: Coefficients are numerical factors multiplied by the variables in a given expression or equation.

Operators: Operators are mathematical symbols that are used to indicate mathematical operations such as addition, subtraction, multiplication and division.

Simplifying Expressions

Simplifying an expression is writing the expression in simplest form by combining like terms and performing operations.

Below is the simple example to demonstrate this:

Simplify the expression: 3x + 4y + 2 + 2x + 9y + 3.

Solution:

For given expression we combine like terms

3x + 2x = 5x

4y + 9y = 13y

2 + 3 = 5

So, the simplified expression is 5x + 13y + 5

Simplifying Equations

We can simplify equations which involves combination of constants and variables with equality sign.

Below is the simple example to demonstrate this:

Example: 3(x+2)-4=2(x+5) Solve this equation

Solution:

First apply distribute property to simplify and expand the equation

3x + 6 - 4 = 2x + 10

Now, combine Like Terms

3x + 2 = 2x + 10

Isolate the Variable

3x - 2x = 10 - 2

x = 8

Examples on Expressions and equations

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

Conclusion

Expressions and equations are fundamental topics in algebra mathematics. These topics play important role in various field and everyday life. It is important to understand how to deal with expression and how to solve equations.

Also Read:

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