Expressions And Equations - GeeksforGeeks (2024)

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.

Table of Content

  • What is an Expression?
    • Types of Expressions
  • What is an Equation?
    • Types of Equation
  • Difference Between Expression and Equation
  • Components of Expressions and Equations
  • Simplifying Expressions
  • Simplifying Equations
  • Examples on 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.

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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

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Difference Between Expression and Equation

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

Characteristics

Expression

Equation

Definition

A combination of numbers, variables, and operators

A statement asserting the equality of two expressions

Contains

No equality sign

An equality sign (=)

Purpose

Represents a value or set of values

Shows that two expressions are equal

Solution

cannot be solved

can be solved

Example

3x + 2

3x + 2 = 7

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

FAQs on Expressions And Equations

What is the difference between expression and equation?

An expression is a combination of numbers, letters, and operations such as addition, subtraction, multiplication and division whereas equation is a mathematical statement that maintain the equality of two expressions.

Can an equation have more than one solution?

Yes, an equation can have more than one solution. Depending on type such as linear equation has one solution, quadratic equation has two solutions and cubic equation has three solutions.

How can we verify that solution of our equation are correct?

We put value of variable again into equation and if both side of equation becomes equal then our equation is correct. This is applicable to every kind of equation such as linear equation, quadratic equation, polynomial equation etc.

What are examples of expressions and equations to show that expressions and equation are somehow related?

Example of expression:

  • 3x + 2
  • 8x + 7

Example of Equation:

3x + 2 = 8x + 7



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FAQs

How do you check equations answers? ›

Substitute the number for the variable in the equation. Simplify the expressions on both sides of the equation. Determine whether the resulting equation is true. If it is true, the number is a solution.

What is the algebraic method of solving linear equations? ›

The algebraic method is a collection of several methods used to solve a pair of linear equations with two variables. The most-commonly used algebraic methods include the substitution method, the elimination method, and the graphing method.

How to solve a pair of linear equations algebraically? ›

This method mainly involves two steps: Step 1: Find the value of one variable, say y in terms of the other variable, i.e., x from either equation, whichever is convenient. Step 2: Substitute this value of y in the other equation, and reduce it to an equation in one variable, i.e., in terms of x, which can be solved.

What is a pair of linear equations in two variables algebraic method? ›

An equation is said to be linear equation in two variables if it is written in the form of ax + by + c=0, where a, b & c are real numbers and the coefficients of x and y, i.e a and b respectively, are not equal to zero. For example, 10x+4y = 3 and -x+5y = 2 are linear equations in two variables.

How to learn algebra easily? ›

Know the order of operations.

One of the trickiest things about solving an algebra equation as a beginner is knowing where to start. Luckily, there's a specific order for solving these problems: first do any math operations in parentheses, then do exponents, then multiply, then divide, then add, and finally subtract.

How to confirm answers algebraically? ›

A solution to an algebra problem is valid if both sides of the equation are still equal when the problem has been worked out with the chosen solution substituted for the variable(s).

What is the easiest way to solve a linear equation? ›

To solve linear equations, find the value of the variable that makes the equation true. Use the inverse of the number that multiplies the variable, and multiply or divide both sides by it. Simplify the result to get the variable value. Check your answer by plugging it back into the equation.

What are the 4 methods of solving linear equations in two variables? ›

To solve a linear equation in two variables, any of the above-mentioned methods can be used i.e. graphical method, elimination method, substitution method, cross multiplication method, matrix method, determinants method.

What are the two algebraic methods of solving systems of equations? ›

Look at two ways to solve systems of linear equations algebraically: substitution and elimination. Look at systems of linear equations graphically to help us understand when systems of linear equations have one solution, no solutions, or infinitely many solutions.

What is the difference between substitution and elimination systems of equations? ›

So, the major difference between the substitution and elimination method is that the substitution method is the process of replacing the variable with a value, whereas the elimination method is the process of removing the variable from the system of linear equations.

How to balance a chemical equation by an algebraic method? ›

Step 1: Write an unbalanced chemical equation for the chemical reaction. Step 2: Assign the variables as the coefficients to each reactant and product. Step 3: Create algebraic equations. Step 4: Assume the value for variable a is 1 and solve the algebraic equations to get the values of each variable.

What is linear pair of linear equation in two variables? ›

A linear equation in two variables is an equation in the form ax+by+c, where a,b, and c are real values and a,b are not equal to zero. We deal with two such equations in a pair of linear equations in two variables. A point on the line representing the equation is the solution of such equations.

How to solve linear equations by elimination method? ›

Key Concepts
  1. Write both equations in standard form. ...
  2. Make the coefficients of one variable opposites. ...
  3. Add the equations resulting from Step 2 to eliminate one variable.
  4. Solve for the remaining variable.
  5. Substitute the solution from Step 4 into one of the original equations. ...
  6. Write the solution as an ordered pair.

How do you check your answer to a system of equations? ›

If you are asked if a point is a solution to an equation, we replace the variables with the given values and see if the 2 sides of the equation are equal (so is a solution), or not equal (so not a solution). A solution to a system of equations means the point must work in both equations in the system.

What is the correct way to check the solved equation? ›

We check a solution to an equation by replacing the variable in the equation with the value of the solution. A solution should result in a true statement when simplified.

How do you check your answer in factoring? ›

You can check your factoring by multiplying them all out to see if you get the original expression. If you do, your factoring is correct; otherwise, you might want to try again.

References

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