Linear Equations in One Variable

Algebra is a branch of mathematics that uses letters and symbols to represent numbers. In previous classes, we learned about simple algebraic expressions. In this chapter, we move one step ahead and study linear equations in one variable, which are very useful in solving real-life problems.

What is a Linear Equation?

An equation is a mathematical statement that shows equality between two expressions using the sign '='. A linear equation in one variable is an equation that can be written in the form:

ax + b = c, where a, b, and c are real numbers and a ≠ 0.

Here, x is the variable whose value we need to find. The degree of the variable is always 1, which is why it is called a linear equation.

Examples of linear equations:

Chapter 2: Linear Equations in One Variable – MCQs
• x + 5 = 12
• 2x − 7 = 9

Non-examples:

• x² + 1 = 0 (degree is 2)

Solution of a Linear Equation

The solution of a linear equation is the value of the variable that makes the equation true. Solving an equation means finding this value.

For example, in the equation x + 5 = 12, subtracting 5 from both sides gives x = 7. Thus, 7 is the solution.

Transposition Method

One of the most common methods of solving linear equations is the method of transposition. In this method, a term is moved from one side of the equation to the other by changing its sign.

For example:

2x + 3 = 11

Transposing 3 to the right side:

2x = 11 − 3

2x = 8

x = 4

Balancing Method

In the balancing method, we perform the same mathematical operation on both sides of the equation to keep it balanced. This method is based on the idea that if equals are added to equals or subtracted from equals, the results are equal.

Equations with Variables on Both Sides

Sometimes, the variable appears on both sides of the equation.

Example:

5x − 3 = 2x + 6

Transposing like terms:

5x − 2x = 6 + 3
3x = 9
x = 3

Reducing Equations to Simpler Form

Some equations may contain brackets or fractions. These equations must first be simplified.

Example:

(x/2) + 3 = 7

Subtract 3 from both sides

x/2 = 4

Multiply both sides by 2:

x = 8

Word Problems Based on Linear Equations

Linear equations are widely used to solve daily life problems involving age, numbers, money, distance, and time. The steps generally followed are:

1. Read the problem carefully

2. Assume the unknown quantity as a variable

3. Form the equation

4. Solve the equation

5. Verify the solution

Example situations include finding ages, prices of items, or numbers with given conditions.

Verification of the Solution

After solving an equation, it is important to verify the solution by substituting the value of the variable back into the original equation. If both sides become equal, the solution is correct.

Importance of Linear Equations

Linear equations help us model and solve real-world problems logically and systematically. They form the foundation for higher algebra and are used in science, economics, and everyday calculations.

Summary

In this chapter, we learned about linear equations in one variable, methods of solving them, simplifying equations, and solving word problems. Mastery of this chapter is essential for understanding advanced algebraic concepts.