Linear Equation in One Variable
2026-02-28 14:06 Diff
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Last updated on October 28, 2025

A linear equation is a simple way to express a mathematical relationship. In a linear equation, ‘x’ represents the unknown quantity. This article explains linear equations in one variable in detail.

What is Linear Equation in One Variable?

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In a linear equation, each variable has a degree of exactly 1.  A linear equation in one variable contains only a variable and results in just one solution.

  • When we draw the linear equation, it makes a straight line.
     
  • Depending on the equation, the graph can be a slanted, horizontal, or vertical line.
  • The general form is ax + b = 0, where x is an unknown variable and a and b are constants. 


For example, adding 7 to an unknown number gives 25. 
In this example, there is only one unknown variable. 
x + 7 = 25 

Difference Between Linear Equation in One Variable vs Non-Linear Equations

Linear Equation

Non-Linear Equation

It forms a straight line when plotted on a graph. 

Non-linear equations create curves or other shapes on graphs.

A linear equation does not have powers or exponents.

These equations include higher powers like squares, fractions, and other powers like x2, y2, etc.

The highest degree of the linear equation is 1.

A non-linear equation has the highest degree of 2 or greater.

Linear equations are used for simple problems like finding speed, making budgets, or calculating total cost based on quantity.

Non-linear equations are used in physics, geometry, etc.

Example: 3x + 2 = 8

Example: x² + y² = 9

How to Solve Linear Equations in One Variable?

A linear equation in one variable has only one variable, and does not include squared terms or similar higher powers. The highest degree of such equations is 1.

Solving a linear equation in one variable can be done using the following methods.

  1. Balancing Method
  2. Transposition Method
  • Balancing Method


In the balancing method, the equation is like a weighing scale; both sides must stay equal.

To solve an equation using balance method, we must do the same thing to both sides:

  1. Add the same number on both sides.
  2. Subtract the same number from both sides
  3. Multiply or divide both sides by the same non-zero number to solve for the variable.
  4. Move the term to the other side by changing its sign.

Example: x - 3 = 7
Add 3 to both sides to eliminate the -3.
\(x - 3 + 3 = 7 + 3\\ x = 10\)

  • Transposition Method


The transposition means moving a term from one side to the other side by changing its sign. 


Example: x + 5 = 12
Move 5 to the other side; it becomes -5.
\(x = 12 - 5\\ x = 7\)
 

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Solving Equations With Variables on One Side

Some equations have variables on one side. To solve these, move the number to the other side, and use the opposite operations to isolate the variable.

Let's understand this using few examples for practice.


Example 1: 2x - 4 = 10

Explanation:

  1. Add 4 to both sides of the equation,
    \(2x - 4 + 4= 10 + 4\\ 2x = 14\)
     
  2. Perform division by 2 on both side.
    \({ 2x \over 2} = {14 \over 2} \\x = 7\)

Example 2: \({2 \over 3} x + {3 \over 6} = 2\)

Explanation:

  1. Multiply both side by the LCM, which is 12
    \(12 \times {2 \over 3} x + 12 \times {3 \over 6} = 12 \times 2 \\ 8x + 6 = 24\)
     
  2. Perform division by 2 on both side.
    \({ 8x \over 2} + {6 \over 2} = {24 \over 2} \\ 4x + 3 = 12\)
     
  3. Move 3 to the other side
    \(4x = 12 -3\\ 4x = 9\)
     
  4. Divide both side by 4
    \({4x \over 4} = {9 \over 4} \\ x = {9 \over 4}\)

Tips and Tricks to Master Linear Equation in One Variable

To understand and effenciently solve linear equations in one variable, here are a few tips and tricks:

  1. Always perform arithmetic operations on both side of the equation.
     
  2. Simplify the equation first to make calculation easy.
     
  3. Don't forget the negative signs.
     
  4. If the equations has fractions, eliminate them by multiplying the entire equation with the LCM.
     
  5. Remember the sign changes when the number moves to another side.

    \(+ \rightarrow - \\ - \rightarrow + \\ \times\rightarrow \div \\ \div\rightarrow \times \)

Parent Tip: Encourage your child to pratice problems from worksheet. Use real life examples to express linear equations to better visualize the linear equations

Common Mistakes and How To Avoid Them in Linear Equations in One Variable

Students make mistakes when solving a linear equation in one variable. Here are some of the common mistakes and the ways to avoid them.

Real Life Applications of Linear Equation in One Variable

Linear equations in one variable are useful when only one unknown quantity needs to be found. Here are some real-life applications of linear equations.

  1. Finance and Budgeting: It is used to track expenses and income, calculate savings, or planning for future spending. If your income is fixed and expenses vary, a linear equation helps you to solve for what you can afford and how much you are left with.
  2. Shopping and Retail: Retailers use linear equations to find discounts, final prices after applying discounts, offers, or adding taxes, and different charges.
  3. Education and Exams: Linear equations help us determine the required scores, averages, or marks needed to improve grades.
  4. Salaries: They are used to calculate the total pay, including bonus, overtime, and deductions while calculating salaries. 

Download Worksheets

Problem 1

Solve 2x + 5 = 13

Okay, lets begin

x = 4
 

Explanation

  1. Subtract 5 from both sides

    \(2x + 5 = 13\\ 2x + 5 - 5 = 13 - 5\\ 2x = 8\)

  2. Divide both sides by 2

    x = 4

Well explained 👍

Problem 2

Solve 3x - 7 = 2x + 1

Okay, lets begin

x = 8
 

Explanation

Move all the x terms to one side, and constants to another.
 

\(3x - 7 = 2x + 1\\ \\ 3x - 2x = 1 + 7\\ x = 8\)
 

Well explained 👍

Problem 3

Solve x/3 + 2 = 5

Okay, lets begin

x = 9
 

Explanation

  1. Subtract 2 from both sides

    \({x \over 3} + 2 = 5\\ {x \over 3} + 2 - 2 = 5 - 2\\ {x \over 3} = 3\)

  2. Multiply both sides by 3

    \(3 \times {x \over 3 } = 3 \times 3 \\x = 9\)

Well explained 👍

Problem 4

Solve 5(x - 2) = 3(x + 4)

Okay, lets begin

x = 11
 

Explanation

  1. Expand both sides

    \(5(x - 2) = 3(x + 4)\\ 5x - 10 = 3x + 12\)

  2. Move x terms to one side

    \(5x - 3x = 12 + 10\\ 2x = 22\)

  3. Divide both sides by 2,

    \({2x \over 2} = {22 \over 2} \\ x = 11\)

Well explained 👍

Problem 5

Solve -4x + 9 = 5

Okay, lets begin

x = 1
 

Explanation

  1. Subtract 9 from both sides

    \(-4x + 9 = 5 \\ -4x = -4\)

  2. Divide by -4

    \({ -4x \over -4} = {-4 \over -4} \\ x = 1\)

Well explained 👍

FAQs on Linear Equation in One Variable

1.How to explain linear equation in one variable to a child?

The linear equation in one variable is expressed as ax + b = 0, where x = variable and a,b are real numbers.

Use examples like, if two cookies cost $2, then cost of 1 cookies is given by 2x = $2, where x is the price of one cookie.
 

2.Why is it important for my child to learn linear equations in one variable?

It is important for your child beacause it builds the foundation of advance concepts of algebra, geometry in maths and can even be used to solve mathmatical problems based on daily life activities.
 

3.What are some mistakes that my child can make in linear equations in one variable?

Some mistakes your children can make are:

  1. Ignoring negative signs.
  2. Not performing operations on both side.
  3. Making calculation errors.
  4. Not simplifying the equations.
     

4.How to explain difference between an equation and an expression to my child?

An expression is a mathematical phrase like 2x + 3, but an equation includes an equal sign and shows a relationship like 2x + 5 = 15.

Explain this using examples, like finding the price of one candy, when the price of 10 candies is known, is an example of equation. Whereas finding the perimeter of a rectangular ground is given using expression 2(l + b).
 

5.My child says linear equation can include powers other than 1. Is this correct?

No, if the variable is squared, cubed, or raised to a higher power, it is not linear. Linear means the highest power is 1.
 

Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.