4x + 6y - 4x + 5y = -32 \\
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Understanding the Equation: 4x + 6y – 4x + 5y = –32
Mathematics often revolves around simplifying complex expressions to uncover meaningful relationships. One such equation—4x + 6y – 4x + 5y = –32—may seem simple, but mastering its breakdown reveals fundamental algebraic principles. In this article, we’ll simplify the equation, solve for key variables, and explore how this equation applies in real-world contexts.
Breaking Down the Equation
The equation starts with:
Let’s simplify step-by-step:
Step 1: Combine Like Terms
Begin by identifying terms with the same variable:
- x-terms: 4x – 4x cancel each other out (resulting in 0)
- y-terms: 6y + 5y = 11y
So the equation simplifies to: 11y = –32
Solving for y
With 11y = –32, divide both sides by 11: y = –32 / 11
This yields a single, precise value for y, streamlining further computations.
What This Means: No x Component
Notably, the x terms cancel completely. This means:
- The equation does not uniquely determine x; x remains a free variable.
- Solutions exist for any value of x, provided y = –32/11.
Practical Implications & Real-World Applications
Equations like 11y = –32出现在多种场景，例如：
1. Financial Models
If x represents time and y represents profit in dollars per unit, the simplified equation shows profit depends entirely on y—a constant rate—while x does not affect outcomes in this model.
2. Physics Problems
In kinematics, if y reflects displacement and x time, the elimination of x implies displacement increases linearly regardless of time span under fixed rate.
3. Engineering Optimization
In constraint-based systems, canceling variables helps identify dependencies, making it easier to allocate resources or find break-even points.
Historical & Conceptual Insight
Algebraic simplification dates back to ancient civilizations, where scholars like Al-Khwarizmi developed systematic approaches to solve equations. Modern teaching emphasizes clearing redundancies—just like simplifying 4x – 4x—to uncover core truths beneath surface formulas.
Step-by-Step Summary
- Simplify: Combine like terms, cancel 4x – 4x → preserves only 11y.
- Equation State: 11y = –32
- Solve: y = –32 / 11
- Interpret: x is arbitrary (any real number); y is fixed.
Conclusion
The equation 4x + 6y – 4x + 5y = –32 simplifies elegantly to 11y = –32, revealing y = –32/11 independent of x. This pattern—eliminating redundant terms to expose key variables—is a cornerstone of algebraic reasoning. Whether in finance, science, or engineering, mastering such simplifications accelerates problem-solving and deepens comprehension.
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Meta Description: Simplify 4x + 6y – 4x + 5y = –32 by combining like terms. Learn how cancellation of 4x reveals y = –32/11 and why x remains free—key concepts in algebraic problem-solving.
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