Simple Equations

What you'll learn

Understand what an equation is and how it differs from an expression
Solve equations using the balancing method (LHS = RHS)
Use the transposition method as a faster approach
Set up and solve equations from word problems

Key concepts

What is an Equation?

An equation is a mathematical statement that two expressions are equal, shown with an = sign.

Term	Meaning	Example
LHS (Left-Hand Side)	Expression to the left of =	2x + 3
RHS (Right-Hand Side)	Expression to the right of =	11
Solution / Root	The value of the variable that makes the equation true	x = 4

Expression vs Equation:

Feature	Expression	Equation
Has = sign	No	Yes
Can be solved	No	Yes
Example	3x + 5	3x + 5 = 14

Types of equations at this level:

Type	Example
One-step	x + 7 = 12
Two-step	3x − 4 = 11
Variable on both sides	2x + 3 = x + 9

Checking a solution: Substitute the value back and verify LHS = RHS.

Check x = 5 in 3x − 4 = 11: LHS = 3(5) − 4 = 15 − 4 = 11 = RHS ✓

Balancing Method

Principle: An equation is like a weighing balance. Whatever you do to one side, you must do the same to the other to keep it balanced.

Operations allowed (done to both sides):

To isolate x	Do this to both sides
x + a = b	Subtract a
x − a = b	Add a
ax = b	Divide by a
x/a = b	Multiply by a

One-step equations:

Example 1: x + 9 = 15 Subtract 9 from both sides: x + 9 − 9 = 15 − 9 x = 6

Example 2: m − 4 = −7 Add 4 to both sides: m = −7 + 4 m = −3

Example 3: 4n = −28 Divide both sides by 4: n = −28/4 n = −7

Example 4: y/5 = 3 Multiply both sides by 5: y = 15

Two-step equations:

Example 5: 2x + 5 = 13 Step 1: Subtract 5 from both sides → 2x = 8 Step 2: Divide both sides by 2 → x = 4 Check: 2(4) + 5 = 13 ✓

Example 6: 3x − 7 = −1 Step 1: Add 7 to both sides → 3x = 6 Step 2: Divide by 3 → x = 2

Variable on both sides:

Example 7: 5x + 3 = 3x + 11 Step 1: Subtract 3x from both sides → 2x + 3 = 11 Step 2: Subtract 3 from both sides → 2x = 8 Step 3: Divide by 2 → x = 4

Transposition Method

Transposition is a shortcut: a term moved from one side to the other changes its sign.

Rules of transposition:

Original position	After crossing =	Sign change
+a on LHS	−a on RHS	+ → −
−a on LHS	+a on RHS	− → +
×a on LHS	÷a on RHS	× → ÷
÷a on LHS	×a on RHS	÷ → ×

Example 1: x + 6 = 10 Transpose +6: x = 10 − 6 = 4

Example 2: 4x − 3 = 9 Transpose −3: 4x = 9 + 3 = 12 Transpose ×4: x = 12/4 = 3

Example 3: (2x + 1)/3 = 5 Transpose ÷3: 2x + 1 = 15 Transpose +1: 2x = 14 Transpose ×2: x = 7

Example 4 — variable both sides: 7y − 2 = 5y + 4 Transpose 5y: 7y − 5y − 2 = 4 Transpose −2: 2y = 6 y = 3

Comparison of methods:

Step	Balancing	Transposition
Speed	Slower, very clear	Faster once practised
Error risk	Lower for beginners	Slightly higher (sign errors)
Best for	Learning the concept	Exam-speed solving

Word Problems

Strategy: Read → identify the unknown → assign a variable → form equation → solve → check → state the answer.

Common word-problem patterns:

Phrase	Equation hint
"A number increased by 5 is 12"	x + 5 = 12
"Twice a number is 18"	2x = 18
"One-third of a number equals 7"	x/3 = 7
"4 more than three times a number is 19"	3x + 4 = 19
"Five less than a number equals 8"	x − 5 = 8

Worked Example 1: The sum of three consecutive integers is 48. Find them. Let the integers be n, n+1, n+2. n + (n+1) + (n+2) = 48 3n + 3 = 48 3n = 45 n = 15 Integers: 15, 16, 17 Check: 15 + 16 + 17 = 48 ✓

Worked Example 2: Arjun is 7 years older than Priya. The sum of their ages is 41. How old is each? Let Priya's age = x, Arjun's age = x + 7 x + (x + 7) = 41 2x + 7 = 41 2x = 34 x = 17 Priya = 17 years, Arjun = 24 years

Worked Example 3: A bag of apples costs ₹15 more than a bag of oranges. Together they cost ₹85. Find the cost of each. Let cost of oranges = ₹x, apples = ₹(x + 15) x + x + 15 = 85 2x = 70 x = 35 Oranges: ₹35, Apples: ₹50

Worked Example 4: The perimeter of a rectangle is 56 cm. Its length is 4 cm more than its breadth. Find the dimensions. Let breadth = b, length = b + 4 2(b + b + 4) = 56 2(2b + 4) = 56 4b + 8 = 56 4b = 48 b = 12 Breadth = 12 cm, Length = 16 cm

Quick check

Solve using the balancing method: 5x + 8 = −12
Solve using transposition: (3y − 1)/4 = 5
Solve: 6m − 3 = 4m + 7
A number multiplied by 4, then decreased by 9 gives 27. Find the number.
The angles of a triangle are in ratio 2 : 3 : 4. Find each angle. (Hint: angles sum = 180°)

Open the Practice tab for graded questions on Simple Equations.