Linear Equations in One Variable

Linear equations in one variable are foundational mathematical concepts that play a vital role in problem-solving across various fields and everyday life. They are used in a wide range of applications, from solving simple arithmetic problems to more complex real-world scenarios. In this chapter, we will explore the concept of linear equations in one variable and understand how to solve them.

What is an Equation?

An equation is a mathematical sentence in which two expressions are equal and it has constants and variables.

Examples:

⇒ 7x − 11 = 3 is an example of an equation which has only one variable (x).

⇒ 7x + 9 = y is an example of an equation which has two variables (x and y).

Linear Equation in One Variable

A linear equation in one variable is an equation that has only one variable which means that the highest power of that variable is 1.

Examples:

⇒ 3x − 2 = 5 is a linear equation in one variable (x).

⇒ 7y + 9 = y is a linear equation in one variable (y).

How to Find the Solution?

The solution of an equation is the value of the variable that makes both sides (LHS and RHS) equal. It is also known as the root of an equation,

Examples:

x + 5 = 8 is satisfied when x = 3. Hence, 3 is the solution of the given equation.
x − 5 = 8 is satisfied when x = 13. Hence, 13 is the solution of the given equation.
5x = 10 is satisfied when x = 2. Hence, 2 is the solution of the given equation.
x/3 = 2 is satisfied when x = 6. Hence, 6 is the solution of the given equation.

Rules of Transposition (Rules for Solving Equations)

When you move a term (a number or variable) from the left-hand side (LHS) of an equation to the right-hand side (RHS), you need to do the opposite operation to keep the equation balanced.

The following steps are transposing a term from LHS to RHS.

1. Addition and Subtraction:

→ If you move something by addition from the LHS to the RHS, you need to subtract it from the RHS.

Example: a + 5 = 2
⇒ a = 2 − 5

→ If you move something by subtraction from the LHS to the RHS, you need to add it to the RHS.

Example: p − 3 = 7
⇒ p = 7 + 3

2. Multiplication and Division:

→ If you move something by multiplication from the LHS to the RHS, you need to divide it on the RHS.

Example: 3y = 11
⇒ y = 11/3

→ If you move something by division from the LHS to the RHS, you need to multiply it on the RHS.

Example: z/3 = 4
⇒ z = 4 × 3

This way, you keep the equation in balance and you can solve it step by step.

How to Solve a Linear Equation?

To solve a linear equation, gather all the variables on one side and all the numbers without the variable on the other side.

Example: The following steps are used to solve a linear equation (x/3) − 5 = x  − 3.

Hence, x = −3 is the solution of the given linear equation.