ICSE Class 6 Maths Chapter 15 Simple Linear Equations

Chapter 15

Simple (Linear) Equations

[Including Word Problems]

Basic Concept

A mathematical statement which shows that two expressions are equal is called an equation.

For example:

If the expressions 3x - 5 and x + 8 are equal, we write 3x - 5 = x + 8, which is an equation.

Similarly:

(i) If the expressions x - 3 and 7 - x are equal, it is written as x - 3 = 7 - x [Which is an equation]

(ii) Seven subtracted from a number (x) equals 4 [A statement]

\(\Rightarrow\) x - 7 = 4 [An equation]

(iii) A certain number (x) multiplied by 4 equals 20 [A statement]

\(\Rightarrow\) 4x = 20 [An equation]

(iv) A number (x) divided by 7 equals 2 [A statement]

\(\Rightarrow\) \(\frac{x}{7}\) = 2 [An equation]

An equation is said to be a linear equation if it contains only one variable (literal) with highest power 1 (one).

Since each of the equations discussed above has only one variable (which is x) with highest power 1, each of these equations is a linear equation.

Solving A Linear Equation

Solving linear equation means finding the value of an unknown algebraic quantity (variable) used in the equation.

For example:

(i) To solve the equation x + 5 = 7 means to find the value of x.

(ii) To solve the equation 3y + 2 = 9 means to find the value of y.

(iii) To solve the equation \(\frac{2a}{3}\) + 4a = 10 means to find the value of a and so on.

Rules For Solving A Linear Equation

Rule 1: The given equation does not change if the same quantity is added on both sides.

e.g. x + 5 = 2 \(\Rightarrow\) x + 5 + 7 = 2 + 7,

3x - 2 = 8 \(\Rightarrow\) 3x - 2 + 4 = 8 + 4, etc.

Rule 2: The given equation does not change if the same quantity is subtracted from both the sides of it.

e.g. x + 5 = 2 \(\Rightarrow\) x + 5 - 7 = 2 - 7,

3x - 2 = 8 \(\Rightarrow\) 3x - 2 - 4 = 8 - 4, etc.

Rule 3: The given equation does not change if each of its terms is multiplied by the same quantity.

e.g. 5x = 2 \(\Rightarrow\) 5x \(\times\) 3 = 2 \(\times\) 3,

\(\frac{3x}{2}\) = 7 \(\Rightarrow\) \(\frac{3x}{2}\) \(\times\) 2 = 7 \(\times\) 2, etc.

Rule 4: The given equation does not change if each of its term is divided by the same non-zero quantity.

e.g. 3x = 5 \(\Rightarrow\) \(\frac{3x}{3}\) = \(\frac{5}{3}\),

7x = 8 \(\Rightarrow\) \(\frac{7x}{4}\) = \(\frac{8}{4}\), etc.

Teacher's Note

Linear equations are the foundation of algebra and appear in real-world problems like calculating costs, distances, and ages - skills essential for everyday financial and practical decisions.

Solving An Equation Of The Form x + a = b

Example 1: Solve: x + 3 = 10

Solution:

x + 3 = 10 \(\Rightarrow\) x + 3 - 3 = 10 - 3 [Rule 2: Subtracting 3 from both the sides]

\(\Rightarrow\) x = 7 (Ans.)

Teacher's Note

When a number is added to the variable, we subtract it from both sides to isolate the variable - this is like removing a jacket to reveal what's underneath.

Solving An Equation Of The Form x - a = b

Example 2: Solve: x - 5 = 2

Solution:

x - 5 = 2 \(\Rightarrow\) x - 5 + 5 = 2 + 5 [Rule 1: Adding 5 on both the sides]

\(\Rightarrow\) x = 7 (Ans.)

Teacher's Note

When a number is subtracted from the variable, we add it to both sides - this reverses the operation to get the original value back.

Solving An Equation Of The Form ax = b

Example 3: Solve: 2x = 6

Solution:

2x = 6 \(\Rightarrow\) \(\frac{2x}{2}\) = \(\frac{6}{2}\) [Rule 4: Dividing each term by 2]

\(\Rightarrow\) x = 3 (Ans.)

Teacher's Note

When a variable is multiplied by a coefficient, dividing both sides by that number isolates the variable - like splitting a recipe ingredient equally among portions.

Solving An Equation Of The Form \(\frac{x}{a}\) = b

Example 4: Solve: \(\frac{b}{2}\) = 5

Solution:

\(\frac{b}{2}\) = 5 \(\Rightarrow\) \(\frac{b}{2}\) \(\times\) 2 = 5 \(\times\) 2 [Rule 3: Multiplying each term by 2]

\(\Rightarrow\) b = 10 (Ans.)

Example 5: Solve: (i) p - 2.5 = 7.3 (ii) x + 3\(\frac{1}{3}\) = 6

Solution:

(i) p - 2.5 = 7.3 \(\Rightarrow\) p - 2.5 + 2.5 = 7.3 + 2.5

\(\Rightarrow\) p = 9.8 (Ans.)

(ii) x + 3\(\frac{1}{3}\) = 6 \(\Rightarrow\) x + \(\frac{10}{3}\) - \(\frac{10}{3}\) = 6 - \(\frac{10}{3}\)

\(\Rightarrow\) x = \(\frac{18-10}{3}\) = \(\frac{8}{3}\) = 2\(\frac{2}{3}\) (Ans.)

Teacher's Note

Mixed numbers and decimals in equations follow the same rules as whole numbers - precision in all forms of numbers is important in cooking, medicine, and engineering.

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