Algebra Play — Class 8 Mathematics (Ganita Prakash)

"Algebra is the language of the universe. With a single equation, x reveals secrets that numbers alone could never tell."

1. About the Chapter

'Algebra Play' is the gateway to modern algebra. This chapter teaches you:

• Linear equations in one variable
• Solving equations systematically
• Translating word problems into equations
• Verifying solutions
• Real-world applications

The skills you build here will be used in every later math class — Class 9 onwards, especially in coordinate geometry, calculus, and physics.

2. What is an Algebraic Equation?

Definition

An algebraic equation is a mathematical statement claiming two expressions are equal, containing one or more variables.

Examples

• 2x + 5 = 13
• 3y − 7 = 2y + 1
• x/4 = 6
• 4(x + 1) = 20

Linear Equation in One Variable

An equation in which:

• Only one variable appears (usually x)
• The variable has degree 1 (no x², x³)
• No fractions with variable in denominator

Examples of linear:

• 3x + 5 = 14
• 2(x − 3) = 8
• x/3 − 4 = 5

NOT linear:

• x² = 25 (quadratic — degree 2)
• 2/x = 5 (variable in denominator)

3. Solving Linear Equations — Basic Principle

The Golden Rule

Whatever you do to one side, do the same to the other.

The equation stays balanced.

Allowed Operations

• Add same number to both sides
• Subtract same number from both sides
• Multiply both sides by same non-zero number
• Divide both sides by same non-zero number

Example

Solve 2x + 5 = 13.

• Subtract 5 from both sides: 2x = 8
• Divide both sides by 2: x = 4

Verify: 2(4) + 5 = 8 + 5 = 13 ✓

4. The Transposition Method (Quick Shortcut)

Idea

When you move a term from one side to the other, change its sign.

• • becomes −
• − becomes +
• × becomes ÷
• ÷ becomes ×

Example

3x − 4 = 11

• Transpose −4 to other side: 3x = 11 + 4 = 15
• Transpose 3 (multiplied to x) to other side as ÷3: x = 15/3 = 5

Why It Works

Transposition is just a shortcut for "+ same number on both sides". It saves writing.

5. Solving Linear Equations — Steps

Standard Procedure

1. Simplify both sides (remove brackets, combine like terms)
2. Collect variable terms on one side, constants on the other
3. Solve for the variable
4. Verify by substituting back

Example with Fractions

Solve (x + 3)/2 = (2x − 1)/3.

Method 1: Cross-multiply

• 3(x + 3) = 2(2x − 1)
• 3x + 9 = 4x − 2
• 9 + 2 = 4x − 3x
• 11 = x
• x = 11

Verify: LHS = (11+3)/2 = 7. RHS = (22−1)/3 = 21/3 = 7. ✓

Example with Brackets

Solve 4(x + 2) = 3(x − 5) + 19.

• 4x + 8 = 3x − 15 + 19
• 4x + 8 = 3x + 4
• 4x − 3x = 4 − 8
• x = −4

Verify: LHS = 4(−4+2) = 4(−2) = −8. RHS = 3(−4−5) + 19 = −27 + 19 = −8 ✓

6. Word Problems — Translating to Equations

The Translation Process

1. Read the problem carefully (often 2-3 times)
2. Identify the unknown (let it be x)
3. Translate the conditions into equations using x
4. Solve the equation
5. Interpret the answer back in the context
6. Verify by checking original conditions

Common Translations

Example

"The sum of three consecutive integers is 72. Find them."

Let smallest = x. Then the three integers are: x, x+1, x+2. Sum: x + (x+1) + (x+2) = 72 3x + 3 = 72 3x = 69 x = 23

Three integers: 23, 24, 25. Verify: 23 + 24 + 25 = 72 ✓

7. Worked Examples

Example 1: Basic

Solve: 7x − 9 = 16.

• 7x = 16 + 9 = 25
• x = 25/7

Example 2: With Brackets

Solve: 5(x − 1) = 3x + 7.

• 5x − 5 = 3x + 7
• 5x − 3x = 7 + 5
• 2x = 12 → x = 6

Example 3: Fractions

Solve: x/3 + x/4 = 7.

• LCM of 3 and 4 = 12
• Multiply both sides by 12: 4x + 3x = 84
• 7x = 84 → x = 12

Example 4: Cross-Multiplication

Solve: (2x − 5)/3 = (x + 1)/2.

• 2(2x − 5) = 3(x + 1)
• 4x − 10 = 3x + 3
• x = 13

Example 5: Age Problem

"A is 8 years older than B. The sum of their ages is 40. Find their ages."

Let B's age = x. Then A's age = x + 8.

• x + (x + 8) = 40
• 2x + 8 = 40
• 2x = 32 → x = 16
• B = 16 years, A = 24 years

Example 6: Money Problem

"A man has ₹100 made of ₹10 and ₹5 coins. He has 11 coins in total. How many of each?"

Let ₹10 coins = x. Then ₹5 coins = 11 − x.

• 10x + 5(11 − x) = 100
• 10x + 55 − 5x = 100
• 5x = 45 → x = 9
• ₹10 coins = 9; ₹5 coins = 2

Example 7: Number Problem

"If 5 is added to thrice a number, the result is 23. Find the number."

Let the number = x.

• 3x + 5 = 23
• 3x = 18 → x = 6

Example 8: Speed Problem

"A car covers a distance at 60 km/h and returns at 40 km/h. The total time is 5 hours. Find the one-way distance."

Let one-way distance = x km.

• Time going = x/60; Time returning = x/40
• x/60 + x/40 = 5
• LCM = 120: 2x + 3x = 600
• 5x = 600 → x = 120 km

8. Common Mistakes

1. Wrong sign in transposition

   • 3x + 5 = 17 → 3x = 17 + 5 (WRONG)
   • 3x + 5 = 17 → 3x = 17 − 5 (CORRECT)

2. Multiplying/dividing only ONE side

   • 2x = 10 → x = 10 (forgot ÷2!)
   • 2x = 10 → x = 5 ✓

3. Not distributing across brackets

   • 2(x + 3) = 6 → x + 3 = 6 (wrong — forgot to distribute or divide both sides by 2)
   • 2(x + 3) = 6 → 2x + 6 = 6 → 2x = 0 → x = 0 ✓

4. Forgetting to verify

   • Always plug answer back into original equation

5. Mistranslating word problems

   • "5 less than x" = x − 5 (NOT 5 − x)
   • "5 more than x" = x + 5

9. Real-World Applications

Personal Finance

• Calculating savings, expenses, taxes
• Discounts and percentages

Science

• Physics formulas: F = ma → a = F/m (algebraic manipulation)
• Chemistry molar calculations

Engineering

• Sizing parts, distances, materials
• Circuit analysis

Computer Science

• Algorithm analysis (variables and equations)
• Database queries (constraints as equations)

Everyday

• Mixing solutions
• Splitting bills evenly
• Calculating time for tasks

10. Tips for Mastery

For Solving

• ALWAYS verify your answer
• Show all steps clearly
• Use transposition for speed

For Word Problems

• READ TWICE before setting up
• Define variable clearly
• Translate one sentence at a time
• After solving, INTERPRET answer in context

Practice

• 10 simple equations daily
• 5 word problems daily
• Mix types (numerical, age, money, speed)

11. Connection to Future Chapters

Class 9

• Linear equations in TWO variables (chapter 4)
• Polynomials (algebraic manipulation)
• Lines and their equations

Class 10

• Quadratic equations (degree 2)
• Coordinate geometry
• Surds and exponents

Class 11-12

• Calculus uses algebraic manipulation everywhere
• Complex numbers
• Trigonometric equations

The algebraic fluency you build NOW will save you years of struggle later.

12. Conclusion

'Algebra Play' is one of the most transformative chapters in your math education. The leap from arithmetic (numbers) to algebra (symbols) is the leap from FOLLOWING instructions to GIVING instructions.

Master:

• The golden rule (do same to both sides)
• Transposition shortcut
• Solving with brackets and fractions
• Word-problem translation
• Verification

Every later science, technology, engineering, and mathematics course depends on these skills. Practise daily, and algebra will become your friend.