Simple Equations Class 7 Notes: Chapter 4

Introduction to Simple Equations

Variables and Expressions

Variable is a quantity that can take any value, its value is not fixed. It is a symbol for a number whose value is unknown yet.

Expressions are formed by performing operations like addition, subtraction, multiplication and division on the variables.

Example: 6x – 3 is an expression in variable x.

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Algebraic Equation

An equation is a condition on a variable such that two expressions in the variable should have equal value.

Example: $8x-8=16$ is an equation.

The value of the variable in an equation for which the equation is satisfied is called the solution of the equation.​

​​​​​​Example: The solution for the equation $2x-3=5$ is $x=4.$
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More about Equations

Mathematical Operations on Expressions

• Addition of variables: $(3x+4z)+(5y+6)$
• Subtraction of variables: $(4x-7y)-(6y+5)$
• Multiplication of variables: $(5xy+6)\times 7x$
• Division of variables: (8xz+5z)/(5x-6y)

Solving an Equation

Solving an equation involves performing the same operations on the expressions on either side of the “=” sign so that the value of the variable is found without disturbing the balance.
Example : Solve $2x+4=10$
Consider $2x+4=10$
$\Rightarrow 2x+4-4=10-4$  [Subtracting 4 from both LHS and RHS] $\Rightarrow 2x=6$
$\Rightarrow 2x/2=6/2$[Dividing both LHS and RHS by 2] $\Rightarrow x=3$

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Methods of Solving an Equation

Method 1: performing the same operations on the expressions on either side of the “=” sign so that the value of the variable is found without disturbing the balance.
Opertions involve Adding, subtracting, multipling or dividing on both sides.

Example: $x+2=6$
Subtract 2 from LHS and RHS
$\Rightarrow$ LHS: $x+2-2=x$
$\Rightarrow$ RHS: $6-2=4$
But LHS = RHS
$\Rightarrow$ x = 4

Method 2: Transposing
It involves moving the terms to one side of the equation to find out the value of the variable.
​​​​​​​When terms move from one side to another they change their sign.
Example: $x+2=6$
Transpose (+2) from LHS to RHS
$\Rightarrow x=6-2$
$\Rightarrow x=4$

Applying Equations

Forming Equation from Solution

Given a solution, many equations can be constructed.

Example:  Given solution: x = 3
Multiply both sides by 4,
$\Rightarrow$ $4x=4\times 3$
Add -5 to both sides,
$\Rightarrow$ $4x-5=12-5$
$\Rightarrow$ $4x-5=7$
Similarly, more equations can be constructed.

Applications (Word problem)

Example: Ram’s father is 3 times as old as his son Ram. After 15 years, he will be twice the age of his son. Form an equation and solve it.
Solution: Let Ram’s age be $x.$
$\Rightarrow$ His father’s age is $3x$.
After 15 years:
$3x+15=2(x+15)$
On solving,
$3x+15=2x+30$
$3x-2x=30-15$
$x=15$
$\therefore$ Ram’s age is 15 and his dad’s age is 45.