Understanding the Equation: From 6² = 130 – 2ab to ab = 47

Mathematics is full of elegant transformations—where simple equations reveal deeper truths. One such example involves the expression 6² = 130 – 2ab, leading step-by-step to the elegant result ab = 47. This method showcases how algebraic manipulation unlocks insights in number theory and problem-solving. Let’s explore the derivation and its implications.

Understanding the Context

The Equation: 6² = 130 – 2ab

We begin with a carefully structured equation:
6² = 130 – 2ab
We know 6² = 36, so substitute to get:
36 = 130 – 2ab

This equation invites us to isolate the product term 2ab, a common algebraic technique used across math problems—from geometry to number theory.

6^2 = 130 - 2ab \Rightarrow 36 = 130 - 2ab \Rightarrow 2ab = 94 \Rightarrow ab = 47

Key Insights

Step 1: Rearranging to Isolate the Product

Subtract 130 from both sides:
36 – 130 = –2ab
–94 = –2ab
Multiply both sides by –1 to simplify:
94 = 2ab

Step 2: Solving for ab

Now divide both sides by 2:
2ab = 94 ⇒ ab = 47

Continue Reading

\(\cos(\theta) = \frac{4}{5}\) (positive in the first quadrant).
Evaluate the sum \(\sum_{k=1}^{5} (2k + 1)\).
Compute each term: when \(k=1\), \(2(1)+1=3\); \(k=2\), \(5\); \(k=3\), \(7\); \(k=4\), \(9\); \(k=5\), \(11\).

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Final Thoughts

This final step reveals a simple yet powerful insight: the product of two variables, a and b, equals 47, given the constraint based on a = 6 and the equation 6² = 130 – 2ab.

Why This Matters: Algebraic Insight and Problem Solving

This derivation highlights key algebraic principles:

  • Substitution: Replacing known values to simplify equations
  • Isolation: Strategically rearranging terms to solve for unknown variables
  • Simplification: Reducing expressions step-by-step for clarity

Such reasoning is essential in solving quadratic expressions, factoring, and balancing equations—skills foundational in algebra, calculus, and advanced math applications.

Real-World Context: Applications of ab = 47

In practical scenarios, knowing that ab = 47 under certain conditions helps in optimization problems, combinatorics, or factoring polynomials. For example:

  • A rectangle with one side fixed at 6 and area derived from ab = 47 implies the other side is 47/6.
  • In algebra, this might appear in expressions like (x – a)(x – b), where a = 6 and ab = 47 leads to a recognizable quadratic pattern.

Conclusion