A weighing scale is used where numbers represent units of weight. The task is to find unknown weights in various configurations of balanced scales.
A sequence of matchstick arrangements is presented (Fig. 7.11, 7.12).
Equations can be solved by substituting different values for the letter-number until LHS = RHS. This is known as the trial and error method.
A key property of equations is that performing the same operation (addition, subtraction, multiplication, or division) on both sides preserves the equality.
Example 1: Given $14593 - 1459 + 145 - 14 + 88 = 13353$, to find $14593 - 1459 + 145 - 14$, subtract 88 from both sides: $13353 - 88$. This works because addition and subtraction are inverse operations.
Example 2: Given $23 \times 41 \times 11 \times 8 \times 7 = 5,80,888$, to find $23 \times 41 \times 11 \times 8$, divide $5,80,888$ by 7. This works because multiplication and division are inverse operations.
Example 3: Given $12345 - 5432 + 135 - 24 - (-67) = 7091$, to find $12345 - 5432 + 135 - 24$, add $(-67)$ to both sides: $7091 + (-67) = 7024$.
Example 4: Given $(35/113) \times 24 \times 14 \times (8/9) = 94080/1017$, to find $(35/113) \times 24 \times 14$, divide both sides by $(8/9)$.
Solving $5x - 4 = 7$ systematically:
Example 5: Solve $11y + (-5) = 61$.
Example 6: Solve $6y + 7 = 4y + 21$.
Observations for systematic solving:
Example 7: Ranjana's tile arrangements. The number of tiles in Step $k$ is $3k+1$. To find which step uses 100 tiles, solve $3k+1=100$.
Example 8: Madhubanti's party snacks. Cost is $₹25$ per plate plus $₹50$ delivery. Total budget $₹500$. If $p$ is the number of plates, the equation is $25p+50=500$.
Example 9: Jahnavi and Sunita saving money. Jahnavi: $₹4000 + ₹650m$ per month. Sunita: $₹5050 + ₹500m$ per month. To find when their savings are equal, solve $4000 + 650m = 5050 + 500m$.
Example 10: Solve $28(x+4) + 300 = 1000$. Multiple methods are shown, including simplifying the LHS first or performing operations on both sides.
Example 11: Riyaz's math trick. If the starting number is $x$, the steps lead to the expression $4x-4$. If the final answer is 24, solve $4x-4=24$ to find $x=7$.
Example 12: Ramesh and Suresh marbles. Total 60 marbles. Ramesh has 30 more than Suresh. If Suresh has $y$ marbles, Ramesh has $y+30$. The equation is $y+(y+30)=60 \Rightarrow 2y+30=60$.
Generating Equations: The reverse process of solving equations is writing equations for a given solution (e.g., $y=5$).
Example 13: A real-life situation for $100x+75=250$ could be: $x$ is the cost per unit of 100 units, and $₹75$ is a fixed charge, with a total cost of $₹250$.
This section presents several solved equations with potential mistakes. The task is to identify and correct these mistakes, then solve the equations correctly.
The use of symbols and solving equations was a significant part of ancient Indian mathematics, known as Bījagaņita (algebra).
The word 'bīja' means seed, implying the unknown is hidden like a seed.
Brahmagupta (628 CE) explained how to work with unknown numbers using letters in his book Brāhmasphuțasiddhānta.
Indian mathematical ideas influenced Al-Khwarizmi (Iraq, 825 CE), whose book 'Hisab al-jabr wal-muqabala' gave rise to the word 'algebra'.
Ancient Indian mathematicians used distinct symbols like yā, kā, nī, pī, lo for different unknowns, similar to modern $x, y$. Known quantities were denoted by rū (rūpa).
Ancient Indian Notation vs. Modern Notation:
Example 16 (Bhāskarāchārya's problem): One man has $₹300$ and 6 horses. Another has 10 horses and a debt of $₹100$. If they are equally rich and a horse costs $₹x$, the equation is $300+6x = 10x-100$. Solving this gives $x=100$.
Aryabhata (499 CE) and Brahmagupta proposed systematic ways to solve problems with single unknowns.
For equations of the form $Ax+B=Cx+D$, Brahmagupta's formula for the solution is $x = (D-B)/(A-C)$.
Example: Solving $650m+4000=500m+5050$ using the formula gives $m = (5050-4000)/(650-500) = 1050/150 = 7$.
Bījagaņita (algebra) is the branch of mathematics that uses letter symbols to solve mathematical problems, generalize patterns, and justify mathematical claims.