Class 7 Maths Chapter 15 Finding the Unknown Visual Infographic | NCF 2023 Ganita Prakash

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✦ NCF 2023 GANITA PRAKASH • GRADE 7 SERIES ✦

Chapter 7: Finding the Unknown

Linear Equations, Balancing Scales, Transposition Rules, and Indian Bījagaṇita Heritage

✨ Explore Interactive TLMs, Worksheets & Visual Notes at: www.akashmaths.online

⚖️ The Balance of Equations

What is an Equation?

A statement showing equality between two expressions (LHS = RHS). A solution makes LHS exactly equal to RHS.

Golden Rule of Balance

Perform the exact same operation (add, subtract, multiply, or divide) on both sides to maintain equality.

Inverse Operations

Addition undoes subtraction; multiplication undoes division (e.g. x − 3 = 5 ⇒ x = 5 + 3).

🔄 Transposition Shortcuts

1. Term Shifting (+ / −):

Moving a term across '=' changes it to its additive inverse (e.g. 2y + 7 = 21 ⇒ 2y = 21 − 7 = 14).

2. Factor Shifting (× / ÷):

A multiplying factor becomes a divisor on the other side (2y = 14 ⇒ y = 14 ÷ 2 = 7). A divisor becomes a multiplier (u/15 = 6 ⇒ u = 6 × 15 = 90).

3. Variables on Both Sides:

Group all variable terms onto one side and all constant numbers onto the other (6y + 7 = 4y + 21 ⇒ 2y = 14).

🏛️ Heritage of Bījagaṇita

The Bīja (Seed) Philosophy:

Bīja means seed. Just as a grand tree is hidden inside a tiny seed, the answer is hidden inside an unknown number.

Ancient Notation:

Unknowns used color initials: yā (yāvat-tāvat), kā (kālaka = black), nī (nīlaka = blue). Constants used rū (rūpa). Negative numbers were marked with a dot above.

Brahmagupta's Linear Formula (628 CE):

For Ax + B = Cx + D ⇒ x = (D − B) / (A − C).

🧠 Real-World Problem Modeling & Historical Classics

• Fixed + Variable Cost Modeling: Total Expense = (Fixed Delivery/Base Charge) + (Rate × Quantity), e.g. 25p + 50 = 500 or 800 + 20k = 2200.

• Bhāskarāchārya's Horse Problem (1150 CE): Man 1 has ₹300 + 6 horses; Man 2 has 10 horses − ₹100 debt. Equal wealth gives 300 + 6x = 10x − 100 ⇒ 1 horse = ₹100.

• Bakhshālī Manuscript Problem (c. 300 CE): 4 people receive x, 2x, 6x, 24x totaling 132 ⇒ 33x = 132 ⇒ First person gets 4.

• Origin of Word "Algebra": Al-Khwarizmi's 825 CE book Hisab al-jabr wal-muqabala (restoring and balancing) gave the name algebra.

⚠️ Common Algebraic Traps

Trap 1: Distributing Outside Multipliers

In 4(4q + 2) = 50, do NOT subtract 2 first! Expand brackets first to 16q + 8 = 50, or divide the whole side by 4 to get 4q + 2 = 12.5.

Trap 2: Sign Flips in Negative Coefficients

When −5x = 3, dividing by −5 yields x = −3/5, NOT −5/3! Always divide by the exact coefficient with its sign intact.

Designed by Akash Srivastva | Mathematics Hub • NCF 2023 Visual Learning Series

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