Title: Solving the Equation –3y(x² + 4xy − 5y²) = −3x²y − 12xy² + 15y³ Step-by-Step

Introduction
Understanding and simplifying algebraic equations is essential for students, researchers, and mathematics enthusiasts alike. One such equation—–3y(x² + 4xy − 5y²) = −3x²y − 12xy² + 15y³—can seem complex at first glance, but with a clear step-by-step breakdown, solving it becomes manageable. This article walks you through the full process of simplifying, expanding, and analyzing the equation to solve it clearly, alongside a practical example and key takeaways.

Understanding the Context

Understanding the Equation
We begin with:
–3y(x² + 4xy − 5y²) = −3x²y − 12xy² + 15y³
This is a bilinear equation in two variables x and y. The left-hand side (LHS) is a product involving y and a quadratic trinomial in x and y, while the right-hand side (RHS) is a cubic polynomial in x and y, including a pure cubic term in y³.

Rewriting the equation in standard form:

→ Move every term to the left-hand side:
–3y(x² + 4xy − 5y²) + 3x²y + 12xy² − 15y³ = 0

Solved: (x^2-4y^2)/xy : (x^2-2xy)/3y = [Math]

Image Gallery

Solved: (x^2-4y^2)/xy : (x^2-2xy)/3y = [Math]
Solved Using | Chegg.com
Q. FACTORIZE− (1) 9x^2 +12xy+4y^2 (2) 25x^2 −20xy+16y^2 (3) 36x^2 +42xy ...
Solved: Factorise: 8x^2+2xy-15y^2 [algebra]
(ii)3x+2y=125x−2y=4 (iv)x+2y−7=02x−y−4=0 | Filo

Key Insights

Step 1: Expand the Left Side
Distribute –3y across the parentheses:
–3y·x² − 3y·4xy + 3y·5y² + 3x²y + 12xy² − 15y³
→ –3x²y − 12xy² + 15y³ + 3x²y + 12xy² − 15y³

Step 2: Combine Like Terms
Group similar terms:

  • x²y: (–3x²y + 3x²y) = 0
  • xy²: (–12xy² + 12xy²) = 0
  • y³: (15y³ – 15y³) = 0

Thus, the entire expression simplifies to:
0 = 0

Continue Reading

Thus, the required matrix is \(oxed{egin{pmatrix} 0 & -2 \ 2 & 0 \end{pmatrix}}\).
A civil engineer analyzing directional stability in solar panel arrays needs to find a vector \(\mathbf{v}\) such that \(\mathbf{v} imes egin{pmatrix} 1 \ 2 \ 3 \end{pmatrix} = egin{pmatrix} 0 \ 0 \ 5 \end{pmatrix}\), where the cross product is defined in 3D. Find a suitable \(\mathbf{v}\).
Let \(\mathbf{v} = egin{pmatrix} x \ y \ z \end{pmatrix}\), \(\mathbf{a} = egin{pmatrix} 1 \ 2 \ 3 \end{pmatrix}\), and \(\mathbf{v} imes \mathbf{a} = egin{pmatrix} 0 \ 0 \ 5 \end{pmatrix}\).

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Final Thoughts

Step 3: Interpret the Result
Since both sides reduce to zero, the equation –3y(x² + 4xy − 5y²) = −3x²y − 12xy² + 15y³ is identically true for all real numbers x and y. In other words, the equation represents a trivial identity — it holds universally.

This means every ordered pair (x, y) satisfies the equation, and thus the expression is an algebraic identity.

Example Substitution (for clarity)
Choose x = 1, y = 2:

  • LHS: –3(2)(1² + 4(1)(2) – 5(2)²) = –6(1 + 8 – 20) = –6(–11) = 66
  • RHS: –3(1)²(2) – 12(1)(2)² + 15(2)³ = –6 – 48 + 120 = 66
    ✅ Both sides equal 66 — confirming the identity.

Why This Matters (Mathematical Insight)
Recognizing identities like this streamlines problem-solving. It avoids unnecessary algebraic steps and clarifies relationships between variables. Identities simplify calculus operations, parametric modeling, and error-checking in applied mathematics.

Key Takeaways

  • Expand and simplify expressions carefully.
  • Combine like terms accurately.
  • Recognizing identical forms (like 0 = 0) indicates an identity.
  • Substituting sample values verifies correctness.
  • This equation is universally true for all real x, y.