Linear Equations in One Variable

Learn how to solve linear equations in one variable using simple steps, examples, transposing method, and student-friendly explanations.

1. Introduction

A linear equation in one variable is an equation that contains only one variable (like \(x\), \(y\), or \(m\)) and the highest power of the variable is 1. These equations can be solved easily using logical steps. Once solved, the equation gives a single value for the variable.

Examples of linear equations in one variable include:

  • \(2x + 5 = 11\)
  • \(7y - 3 = 18\)
  • \(3m = 21\)

2. Standard Form of a One-Variable Linear Equation

The standard form of a linear equation in one variable is:

\(ax + b = 0\)

Here:

Any equation that can be converted to this form is a one-variable linear equation.

2.1. Examples of Standard Form

  • \(2x + 5 = 0\)
  • \(7x - 3 = 0\)
  • \(3p + 10 = 0\)

3. Steps to Solve Linear Equations

To solve a linear equation, we isolate the variable by performing the same operations on both sides of the equation. This keeps the equation balanced.

Important idea: Whatever operation you do on one side must be done on the other side.

3.1. Using the Transposing Method

Transposing means taking a term to the other side and changing its sign.

For example:

\(2x + 5 = 11\)

  • Take \(+5\) to the other side: \(2x = 11 - 5\)
  • \(2x = 6\)
  • \(x = 3\)

3.2. Solving Equations with Fractions

Clear the denominator first by multiplying both sides by the LCM.

Example:

\(\dfrac{x}{3} + 2 = 5\)

  • Subtract 2: \(\dfrac{x}{3} = 3\)
  • Multiply both sides by 3: \(x = 9\)

3.3. Equations with Variables on Both Sides

Move all variable terms to one side and constants to the other.

Example:

\(5x - 3 = 2x + 9\)

  • Move \(2x\): \(5x - 2x = 9 + 3\)
  • \(3x = 12\)
  • \(x = 4\)

4. Worked Examples

Let us solve some equations step by step.

4.1. Basic Examples

1. Solve \(3x = 12\)

  • Divide both sides by 3: \(x = 4\)

2. Solve \(x - 7 = 10\)

  • Add 7: \(x = 17\)

4.2. Medium Difficulty Examples

1. Solve \(4x + 9 = 25\)

  • Subtract 9: \(4x = 16\)
  • Divide by 4: \(x = 4\)

2. Solve \(2(x - 3) = 8\)

  • Expand: \(2x - 6 = 8\)
  • Add 6: \(2x = 14\)
  • Divide by 2: \(x = 7\)

4.3. Word-Problem Examples

Example: A number increased by 7 gives 19. Find the number.

Let the number be \(x\).

Equation: \(x + 7 = 19\)

  • \(x = 19 - 7 = 12\)

5. Common Mistakes

  • Changing signs incorrectly while transposing.
  • Not performing the same operation on both sides.
  • Forgetting to simplify after each step.
  • Incorrectly expanding brackets.

6. Quick Practice

Solve the following equations:

  1. \(3x + 4 = 19\)
  2. \(7y - 5 = 23\)
  3. \(5p + 9 = 34\)
  4. \(4m - 2 = 2m + 10\)

7. Summary

  • A one-variable linear equation has the form \(ax + b = 0\).
  • To solve it, isolate the variable using balancing or transposing.
  • Clear fractions by multiplying both sides by the LCM.
  • Move all variable terms to one side and constants to the other when necessary.
  • Always check your solution by substituting it back.