Linear Equation in One Variable

Introduction to Linear Equation in One Variable

A linear equation in one variable is a simple equation with just one unknown number, called a variable. The variable is usually written as x, y, or z, and the power is always 1. These equations are called "linear" because if they are drawn on the graph, they form a straight line. Examples are x + 4 = 9 or 2x - 3 = 7.

Linear equations in one variable are very useful in mathematics because they help us solve problems where we need to find an unknown number. By learning the rules and steps carefully, we can easily solve such equations and apply them in real-life situations such as age problems, money problems, or measurement problems.

Table of Content

• Definition
• Standard Form
• Steps for Solving
• Examples
• Application
• Word Problems
• Practice Questions
• Conclusion
• FAQs on Linear Equation in One Variable
  
  

Definition

A linear equation in one variable is a type of algebraic equation that contains one unknown variable, and the highest power of the variable is 1. These equations show a straight-line relationship between the variable and the constants.

The main goal of solving such an equation is to find the value of the variable that makes the equation true. To do this, we also use basic math operations such as addition, subtraction, multiplication, and division. For example, in x + 4 = 9, we subtract 4 from both sides and get x = 5. Similarly, in 2x - 3 = 7, we add 3 to both sides, then divide by 2, and get x = 5.

It is called “linear” because if you draw it on a graph, it makes a straight line.

Standard Form

The common way to write it is:

$ax + b = 0 \left( a\neq 0 \right)$

Here:

• x is a variable (unknown value)

• a coefficient is a non-zero constant.)

• b is the constant term.

Example:

• 3x - 6 = 0

• X + 4 = 10

Structure Overview Table:

Steps for Solving

When we solve an equation that contains only one variable, we follow a few simple steps. These steps help us find the value of the variable correctly.

Steps to solve:

Step 1: If there are fractions in the equation, remove them using LCM.

Step 2: Simplify both sides of the equation by doing basic operations.

Step 3: Move all variables to one side and numbers to the other side. Remember that when you move, the sign of numbers changes.

Step 4: Solve for the variable and check your answer by putting it back in the original equation.

Example of a solution 

Consider the equation 3x + 7 = 16

• Step 1: Keep the variable term on one side. Subtract 7 from both sides:
  3x + 7 – 7 = 16 – 7
  ⇒ 3x = 9

• Step 2: Now, divide both sides by 3:
  3x ÷ 3 = 9 ÷ 3
  ⇒ x = 3

• Step 3: Verify the answer. Put x = 3 back in the original equation:
  3(3) + 7 = 16
  ⇒ 9 + 7 = 16
  ⇒ 16 = 16

• So, the solution is x = 3.
  
  

Examples

1. Solve for x: 2x – 5 = 3

Solution:

• Add 5 to both sides:

• 2x – 5 + 5 = 3 + 5

• 2x = 8

• Divide both sides by 2:

• 2x ÷ 2 = 8 ÷ 2

• x = 4

• Answer: x = 4

2. Solve for y: 5y + 7 = 22

Solution:

• Subtract 7 from both sides:

• 5y + 7 – 7 = 22 – 7

• 5y = 15

• Divide both sides by 5:

• 5y ÷ 5 = 15 ÷ 5

• y = 3

• Answer: y = 3

3. Solve for m: 6m – 10 = 8

Solution:

• Add 10 to both sides:

  • 6m – 10 + 10 = 8 + 10

  • 6m = 18

• Divide both sides by 6:

  • 6m ÷ 6 = 18 ÷ 6

  • m = 3

• Answer: m = 3

Application

Linear equations are not just class exercises; they are useful in many real situations. Some common applications include:

• Age problems: Find out the age of a person when given relations between ages.

• Money transactions: Calculation of prices, discounts, or amounts after spending or savings.

• Measurement problems: Solution for unknown length, width, or height of simple shapes.

• Daily Plan: Dividing time for activities such as studies, travel, or play when the total time is resolved.

Word Problems

Example: The length of the legs of an isosceles triangle is 3 metres more than its base. If the perimeter of the triangle is 39 metres, find the lengths of all sides.

Solution:

• Let the base of the triangle be x metres.

• Then, each of the legs will be x + 3 metres.

• The perimeter of a triangle is the sum of all three sides. So we can write the equation: x + (x + 3) + (x + 3) = 39

• Simplify the equation:

• x + x + 3 + x + 3 = 39

• 3x + 6 = 39

• Subtract 6 from both sides:

• 3x = 39 – 6

• 3x = 33

• Divide both sides by 3:

• x = 33 ÷ 3

• x = 11

• So, the base is 11 metres, and each leg is 11 + 3 = 14 metres.
  
  

Practice Questions

1. Solve 8x – 5 = 19

2. Find two consecutive multiples of 7 if their sum is 91.

3. Check if x = –2 is a solution of 6x + 9 = 21 – 4x.

Conclusion

Linear equations in one variable are the most important fundamental in algebra. They are easy to understand and solve because they only have one unknown number. By following the step-by-step process of simplifying, moving terms, and solving, we can easily get the right value of the variable. Learning these equations creates a strong foundation for more advanced topics in mathematics, like linear equations in two variables and beyond. Mastering this concept helps to solve practical problems quickly and correctly.

FAQs on Linear Equation in One Variable

1. What is a linear equation in one variable with examples?

A linear equation in one variable is an equation with only one unknown variable, and the highest power of the variable is 1. It is written as:

$𝐴 𝑥 + 𝐵= 0$

Where A and B are real numbers and $\left( a\neq 0 \right)$.

Example: 9x + 78 = 18

2. What is a linear equation in one variable expression?

A single variable expression is a mathematical expression where only one variable is in it (e.g., 2x + 3). When we put an equal sign (=) in it, there is a linear equation in a variable.

Example: Expressions → 2x + 3, Equation → 2x + 3 = 0

3. How to solve linear equations with 1 variable?

To solve:

• Simplify both sides (remove brackets, mix as terms).

• Move all the variable terms to one side and the constant to the other side.

• Use addition/subtraction to isolate the variable.

• Divide or multiply to find the value of the variable.
  Example: Loose 2x + 5 = 11
  → 2x = 11 - 5
  → 2x = 6
  → x = 3

4. Is 2x + 3 a linear equation in one variable?

Number 2x + 3 is just an expression because it has no equal sign.

If we write it as 2x + 3 = 0, it becomes a linear equation in a variable.

5. What are 5 examples of linear equations?

Here are 5 examples:

• $2x - 3 = 0$

• $5x + 7 = 12$

• $\frac{x}{2} = 3$

• $9x - 4 = 0$

• $4x + 11 = 27$