Linear Equations in one Variable Chapter Notes | Mathematics Class 8 ICSE PDF Download

Introduction

Linear equations in one variable are fundamental algebraic expressions that involve a single variable with a power of one. This chapter introduces the concept of equations, focusing on linear equations, their properties, and methods to solve them. It also covers how to apply these equations to solve real-world problems by forming equations based on given conditions. By understanding the steps to manipulate and solve these equations, students can find the value of the unknown variable, which is the key to mastering this topic.

Equation

• ​An equation is a mathematical statement that shows two algebraic expressions are equal.
• It consists of two sides connected by an equal sign.
• Both sides must have the same value.
• Example: Consider the equation 2x + 5 = 11.
  Here, 2x + 5 (left side) equals 11 (right side).
  Step 1: Subtract 5 from both sides: 2x + 5 - 5 = 11 - 5 → 2x = 6.

  Step 2: Divide both sides by 2: 2x ÷ 2 = 6 ÷ 2 → x = 3.
  So, x = 3 satisfies the equation.

Linear Equation

• ​A linear equation involves only one variable with the highest power of one.
• It can be written in the form ax + b = 0, where a and b are constants, and a ≠ 0.
• It represents a straight line when graphed.
• Example: Solve 4x - 8 = 0.

  Step 1: Add 8 to both sides: 4x - 8 + 8 = 0 + 8 → 4x = 8.
  Step 2: Divide both sides by 4: 4x ÷ 4 = 8 ÷ 4 → x = 2.
  So, x = 2 is the solution.

Solving an Equation

• ​To solve an equation means to find the value of the variable that makes the equation true.
• The solution is the value that satisfies both sides of the equation.
• It involves isolating the variable using algebraic operations.
• Example: Solve 3y + 7 = 16.

  Step 1: Subtract 7 from both sides: 3y + 7 - 7 = 16 - 7 → 3y = 9.
  Step 2: Divide both sides by 3: 3y ÷ 3 = 9 ÷ 3 → y = 3.
  So, y = 3 is the solution.

​Root of a Linear Equation

• ​A linear equation has only one solution, called its root.
• The root is the value of the variable that satisfies the equation.
• It is unique for linear equations in one variable.
• Example: Find the root of 5z - 10 = 0.

  Step 1: Add 10 to both sides: 5z - 10 + 10 = 0 + 10 → 5z = 10.
  Step 2: Divide both sides by 5: 5z ÷ 5 = 10 ÷ 5 → z = 2.
  The root is z = 2.

Properties of Equations

• ​An equation remains unchanged when:
• The same number is added to both sides.
• The same number is subtracted from both sides.
• Both sides are multiplied by the same non-zero number.
• Both sides are divided by the same non-zero number.
• Example: Solve 2x - 3 = 7 using properties.

  Step 1: Add 3 to both sides: 2x - 3 + 3 = 7 + 3 → 2x = 10.
  Step 2: Divide both sides by 2: 2x ÷ 2 = 10 ÷ 2 → x = 5.
  So, x = 5 is the solution.

To Solve Problems Based on Linear Equations

Steps to Solve Word Problems

• Read the problem carefully to identify what is given and what needs to be found.
• Represent the unknown quantity with a variable (e.g., x, y, z).
• Form an equation based on the relationship between the given quantities and the unknown.
• Solve the equation to find the value of the unknown.
• Example: A number is such that twice the number is 10 more than the number itself. Find the number.

  Step 1: Let the number be x.
  Step 2: Form the equation: 2x = x + 10.
  Step 3: Subtract x from both sides: 2x - x = x + 10 - x → x = 10.
  So, the number is 10.

Consecutive Numbers

• Consecutive numbers: Represent as x, x + 1, x + 2, etc.
• Consecutive even numbers: Represent as x, x + 2, x + 4, where x is even.
• Consecutive odd numbers: Represent as x, x + 2, x + 4, where x is odd.
• Consecutive multiples of 3: Represent as x, x + 3, x + 6, where x is a multiple of 3.
• Example: The sum of two consecutive odd numbers is 36. Find the numbers.

  Step 1: Let the first odd number be x, then the next is x + 2.
  Step 2: Form the equation: x + (x + 2) = 36.
  Step 3: Simplify: 2x + 2 = 36.
  Step 4: Subtract 2 from both sides: 2x = 34.
  Step 5: Divide by 2: x = 17.
  So, the numbers are 17 and 19.

Example: A rectangle is 8 cm long and 5 cm wide. Its perimeter is doubled when each of its sides is increased by x cm. Form an equation in x and find the new length of the rectangle.

Solution :
Since, length of the rectangle = 8 cm and its width = 5 cm
Its perimeter = 2(length + width)
= 2(8 + 5) cm = 26 cm

On increasing each of its sides by x cm,
its new length = (8 + x) cm
and, new width = (5 + x) cm
Its new perimeter = 2(8 + x + 5 + x) cm
= (26 + 4x) cm

Given : new perimeter = 2 times the original perimeter
26 + 4x = 2 × 26
4x = 52 - 26 = 26
x = 26/4 = 6.5 cm
And, the new length of the rectangle = (8 + x) cm
= (8 + 6.5) cm = 14.5 cm. (Ans.)

Example: A man is 24 years older than his son. In 2 years, his age will be twice the age of his son. Find their present ages.

Solution :
Let the present age of the son be x years
Therefore Present age of the father = (x + 24) years

In 2 years :
The man's age will be (x + 24) + 2 = (x + 26) years
and son's age will be x + 2 years

According to the problem : x + 26 = 2(x + 2)
On solving we get : x + 26 = 2x + 4
x = 22

Therefore Present age of the man = x + 24 = 22 + 24 = 46 years
and, Present age of the son = x = 22 years. (Ans.)