Understanding the Problem and Solution Method

The question asks us to identify the method used in the provided solution to prove an algebraic relationship based on a given proportion. The given proportion is \(\frac{a}{b} = \frac{c}{d}\), and the solution shows steps leading to the conclusion \(d(a - 2ab) = b(c - 2ad)\).

Let's look at the steps provided in the solution:

1. It starts with the known fact: \(\frac{a}{b} = \frac{c}{d}\).
2. It then performs an operation on both sides: subtracts \(2a\) from both sides, resulting in \(\frac{a}{b} - 2a = \frac{c}{d} - 2a\).
3. The final step shown is \(d(a - 2ab) = b(c - 2ad)\). While the intermediate steps to get from step 2 to step 3 are not explicitly shown, they involve algebraic manipulations like multiplying by \(bd\) on both sides.

The key observation is that the solution begins with a known premise (\(\frac{a}{b} = \frac{c}{d}\)) and logically derives the desired conclusion \(d(a - 2ab) = b(c - 2ad)\) through a series of steps.

Exploring Mathematical Methods for Proof

There are different methods used in mathematics to solve problems or prove statements. Let's briefly look at the options provided:

• Analytic Method: This method starts by assuming the conclusion or the statement to be proved is true. Then, it works backward, step-by-step, to arrive at a known truth, definition, or axiom. It's like dissecting the problem to find its fundamental components.
• Synthetic Method: This method starts with known facts, definitions, axioms, or previously proved theorems. It then proceeds step-by-step using logical deduction to construct the proof and arrive at the desired conclusion. It's like building up the solution from known parts.
• Inductive Method: This method involves observing specific instances or examples and then inferring a general rule or conclusion. While useful in generating hypotheses, it's not a formal proof method for establishing the absolute truth of a mathematical statement for all cases (except for mathematical induction, which is a specific proof technique).
• Heuristic Method: This refers to a problem-solving approach that uses practical techniques or rules of thumb that are not guaranteed to be optimal or correct but are sufficient for the immediate goals. It's about finding a solution, possibly through trial and error, rather than providing a rigorous proof.

Identifying the Method Used in the Solution

In the given solution, the process starts from the initial premise \(\frac{a}{b} = \frac{c}{d}\), which is given as true. The solution then applies algebraic operations (subtracting \(2a\), multiplying by \(bd\), etc.) to transform this initial statement until it reaches the expression \(d(a - 2ab) = b(c - 2ad)\). This is a clear example of working forward from a known truth to the desired result.

Therefore, the method used is the Synthetic Method because it synthesizes, or builds up, the conclusion from the initial given condition through a sequence of logical steps.

Conclusion

The solution provided starts with the given proportion and proceeds through algebraic manipulations to arrive at the required expression. This step-by-step derivation from a known fact to the conclusion is characteristic of the Synthetic Method.