CBSE Notes Class 7 Maths Chapter 4 Simple Equations

In Chapter 4: Simple Equations, students are introduced to the concept of equations—a fundamental building block of algebra. This chapter lays the foundation for solving real-life problems using algebraic expressions and equations. It teaches how to form and solve simple equations, understand the equality between two mathematical expressions, and use equations as a tool for problem-solving.

This topic not only sharpens logical thinking but also prepares students for higher-level algebraic concepts they will encounter in the future.

1.0Introduction to Simple Equations

1. Variables: In a mathematical equation, a variable is a letter or alphabet used in place of an unspecified number in expressions, equations or formulas. That is, in the given expression a variable act as a placeholder for the unknown number. Generally, a single letter is used to represent a variable. Ex- x, y, z, t etc. 
2. Expression: An expression or algebraic expression is a mathematical statement made up of numbers, variables, and arithmetic operations (such as +, –, ×, ÷). For example, in the expression 4m + 5, the terms are 4m and 5, with m being the variable. These terms are connected by the arithmetic operation +. 
3. Equations: An equation is a mathematical statement that shows the equality between two expressions and includes one or more unknown variables. 

2.0Linear Equations or Simple Equations 

An equation involving only a linear polynomial is called a linear equation.

Example: 4x + 2 = –2 

$\frac{5}{4}x+9=\frac{1}{2}$

Solution of Linear Equation: The value of a variable that satisfies the equation and makes both sides equal is called the solution or root of the equation.

Example: 

The solution of the equation 9 + x = 13 is 4. In other words, the value of x, called the variable, which satisfies the given equation is called the solution or root of the equation.

3.0Balancing an Equation  

To solve an equation, both sides must remain balanced. Follow these four key rules:  

1. Addition: Adding the same number to both sides keeps the equation balanced.  
2. Subtraction: Subtracting the same number from both sides maintains equality.  
3. Multiplication: Multiply both sides by the same number to preserve balance.  
4. Division: To preserve equality, divide both sides of the equation by the same non-zero number.  

4.0Transposing Method

In this method, we transpose numbers from one side of the equation to the other, moving all terms with variables to one side and constants to the other. When transposing:  

• A positive number becomes negative, and a negative number becomes positive.  
• Similarly, multiplication changes to division, and division changes to multiplication.

Transposition helps in isolating the variable and solving simple equations efficiently.

5.0From Solution to Equation 

We cannot only solve an equation, but also can make equations by following the reverse path. Also, given an equation, we can get one solution but with the given solution we can make many equations.

1. What’s a Solution and a Variable?  

A solution is the value that satisfies an equation, while a variable (like x or y) is a letter that stands for an unknown number. For example, if we know that x = 5, we can create different equations, like 2x + 3 = 13.

2. Turning Word Problems into Equations:  

When you read a word problem, look for important details. Identify what the problem is asking for and what the unknowns are. For instance, if the problem says, “A number increased by 4 equals 10,” you can let x represent that number and write it as:  x + 4 = 10

3. Balancing the Equation:  

To find the value of x, you need to keep the equation balanced. This means doing the same thing to both sides. For example, if you have x + 4 = 10, you can subtract 4 from both sides to isolate x:  

x + 4 – 4 = 10 – 4  

This simplifies to x = 6.

6.0Solved Examples on Simple Equations

Example 1: Check whether the value given in the bracket is a solution (root) of the given equation or not?

y – 11 = 4 (at y = 16)

Solution:

Substituting y = 16 in the given equation, we get 

L.H.S. = 16 – 11 = 5 ≠ R.H.S.

Example 2: Solve the following equations without transposing:  2x + 14 = 26

Solution: 

2x + 14 = 26 

Subtract 14 from both sides

2x + 14 – 14 = 26 – 14 

2x = 12 

Divide both the sides by 2 

$\frac{2x}{2}=\frac{12}{2}$

x = 6

Example 3: Solve the equation 3 + 2(p – 7) = 9

Solution: 

3 + 2(p – 7) = 9 

2(p – 7) = 9 – 3 

2(p – 7) = 6 

$P-7=\frac{6}{2}$

p – 7 = 3 

p = 3 + 7 

p = 10

Example 4: Solve $x-(2x-\frac{3x-4}{7})=\frac{4x-27}{3}-3$

Solution: 

We have:

$x-(2x-\frac{3x-4}{7})=\frac{4x-27}{3}-3$

Removing the brackets, we get

$x-2x=\frac{3x-4}{7}=\frac{4x-27}{3}-3$

$-x+\frac{3x-4}{7}=\frac{4x-27}{3}-3$

Multiplying both sides by 21, the LCM of 7 and 3, we get 

–21x + 3(3x – 4) = 7(4x – 27) – 63 

 –21x + 9x – 12 = 28x – 189 – 63 

–12x –12 = 28x – 252 

–12x – 28x = –252 + 12 [by transposing 28x on LHS & –12 on RHS] 

 –40x = –240 ⇒ x = 6 [on dividing both sides by –40]. 

 x = 6 is a solution of the given equation.

Example 5: If x = 8, then find the equation.

Solution: 

We have, x = 8 

x + 5 = 8 + 5 (Adding 5 to both sides) 

x + 5 = 13 

Hence, x + 5 = 13 is an equation for x = 8 

Another way, 

x = 8 

x × 4 = 8 × 4 (Multiply 4 to both sides) 

4x = 32 

Hence, 4x = 32 is also an equation for x = 8. 

In this way, we can find as many equations as we want for the solution x = 8.