Linear Equations in One Variable Chapter Notes | Mathematics Class 6 ICSE PDF Download

Introduction

Imagine you're on a treasure hunt, and the map gives you clues in the form of equations! Linear equations in one variable are like simple puzzles where you need to find the value of a single unknown, such as x or y. These equations are called "linear" because they represent a straight line when graphed, and they only involve the variable raised to the power of one. This chapter is all about turning words into math, moving terms around, and solving these puzzles step-by-step to uncover the hidden number. Let’s dive into this exciting world of equations and learn how to crack them!

To Write a Given Statement in Algebraic Form

• Convert a word statement into a mathematical expression by using letters for unknowns and symbols for operations.
• Identify key phrases like "added to," "subtracted from," "multiplied by," or "divided by" to assign the correct operations.
• Use a variable (like x or y) to represent the unknown number.
• Ensure the equation reflects the relationship described in the statement.

To Write a Statement for a Given Algebraic Form

• Translate a mathematical equation into a word statement by interpreting the symbols and variables.
• Replace the variable with words like "a number" or the variable’s name.
• Describe operations like + (added to), - (subtracted from), × (multiplied by), or ÷ (divided by) in words.
• Ensure the statement clearly conveys the equation’s meaning.

Linear Equations in One Variable

• An equation is a statement showing that two expressions are equal, with an equals sign (=) between them.
• A linear equation has variables with the highest power of one (no x² or higher).
• A linear equation in one variable involves only one unknown, like x or n.
• Examples include 2n + 3 = 7 or x + 5 = 4x.

Properties of Equations

• Addition Property: Adding the same number to both sides keeps the equation balanced.
  Formula: If x = y, then x + z = y + z.
• Subtraction Property: Subtracting the same number from both sides keeps the equation balanced.
  Formula: If x = y, then x - z = y - z.
• Multiplication Property: Multiplying both sides by the same number keeps the equation balanced.
  Formula: If x = y, then x × z = y × z.
• Division Property: Dividing both sides by the same non-zero number keeps the equation balanced.
  Formula: If x = y and z ≠ 0, then x ÷ z = y ÷ z.

Transposition of Terms of an Equation

• Move a term from one side of the equation to the other by changing its sign (e.g., + becomes -, × becomes ÷).
• Addition becomes subtraction, subtraction becomes addition, multiplication becomes division, and division becomes multiplication when transposing.
• This simplifies the equation by grouping similar terms.

• The solution is the value of the variable that makes the equation true.
• Substitute the solution into the equation to verify it satisfies both sides.

Solving Linear Equations in One Variable

• Move all terms with the variable to one side using transposition.
• Move all constant terms to the other side.
• Simplify the equation by combining like terms.
• Divide both sides by the coefficient of the variable to find its value.

Verification of the Solution

• After solving, check the solution by substituting the variable’s value into the original equation.
• Calculate the left-hand side (LHS) and right-hand side (RHS) separately.
• If LHS equals RHS, the solution is correct.
• This step ensures no mistakes were made during solving.

Word Problems

• Read the problem carefully to understand the situation.
• Assign a variable to the unknown quantity.
• Form an equation based on the given conditions.
• Solve the equation using the steps for linear equations.
• Verify the solution by checking if it fits the problem’s conditions.

Solved Examples