Chapter 15 — Finding the Unknown

It introduces algebraic equations for Class 7 students. Starting from weighing-scale and matchstick puzzles, it teaches how to represent an unknown quantity with a letter-number, frame an equation, and solve it systematically using inverse operations.

A statement of equality between two algebraic expressions is called an equation. Examples from the chapter include 2n + 1 = 99, 3x + 4 = 7, and 2z + 4 = 5z – 14.

Solving an equation means finding the value(s) of the letter-number for which the equality holds — that is, for which the Left Hand Side (LHS) of the equation equals the Right Hand Side (RHS).

In trial and error, you substitute different values in place of the unknown and check which value makes LHS = RHS. For example, solving 2n + 1 = 99 by trying n = 5, 10, 30, 40, 49 in turn until LHS equals 99. The chapter notes this method can be inefficient.

Perform the same operation on both sides to isolate the unknown. For 5x – 4 = 7, first add 4 to both sides to get 5x = 11, then divide both sides by 5 to get x = 11/5. You can verify by substituting back into the original equation.

Bring the unknown terms to one side by subtracting the smaller unknown term from both sides. For 6y + 7 = 4y + 21, subtract 4y from both sides to get 2y + 7 = 21, then subtract 7 to get 2y = 14, giving y = 7.

The chapter states: (a) a term added or subtracted on one side that is removed must appear as its additive inverse on the other side; (b) if a factor is removed from one side, the other side must be divided by that factor; (c) if a divisor is removed, the other side must be multiplied by that divisor — for example u/15 = 6 becomes u = 6 × 15 = 90.

The matchstick sequence has 2n + 1 sticks at position n (position 1 has 3, position 2 has 5, etc.). To find which position uses exactly 99 sticks, frame the equation 2n + 1 = 99 and solve it; the answer is n = 49. Whether 200 sticks are possible is posed as a discussion question.

Bījagaṇita is the ancient Indian term for algebra. The word bīja means seed — just as a tree is hidden inside a seed, the answer is hidden inside an unknown. Brahmagupta outlined a systematic method to solve equations in Chapter 18 of Brāhmasphuṭasiddhānta (628 CE). Around 825 CE Al-Khwarizmi's book Hisab al-jabr wal-muqabala was influenced by these Indian ideas; when translated into Latin in the 12th century, the word al-jabr became our word 'algebra'.

They used symbols like yā, kā, nī, pī, lo for different unknowns. yā was short for yāvat-tāvat ('as much as needed'); kā and nī came from the first letters of colour names — kālaka (black) and nīlaka (blue). Known quantities were denoted by rū (rūpa). For example, modern 2x + 1 was written as 'yā 2 rū 1'.

Brahmagupta's formula gives x = (D – B) / (A – C). For example, to solve 650m + 4000 = 500m + 5050, calculate m = (5050 – 4000) / (650 – 500) = 1050 / 150 = 7.

Substitute the solution back into the original equation and verify that LHS equals RHS. For example, for x = 11/5 in the equation 5x – 4 = 7, compute LHS = 5 × (11/5) – 4 = 11 – 4 = 7, which equals RHS.

Section 7.3 presents nine solved equations, some containing errors. Students must go through each step, identify whether it is correct, describe any mistake, correct it, and solve the equation properly. This trains careful, step-by-step algebraic reasoning.

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