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Adding System Equations: Resultant Equation

Combining equations within a system involves summing their left-hand sides and their right-hand sides separately to create a new equation. For instance, given the system x + y = 5 and 2x – y = 1, summing the left sides (x + y) + (2x – y) yields 3x, and summing the right sides 5 + 1 results in 6. The resulting equation is 3x = 6.

This process is fundamental in solving systems of equations. It allows for the elimination of variables, simplifying the system and enabling the determination of solutions. This technique appears across diverse mathematical fields, from basic algebra to linear programming and beyond, proving essential for solving complex problems. Its historical roots lie within the development of algebraic methods, crucial for advancements in various scientific and engineering disciplines.

Sum of Two Equations: Result & Calculation

Combining two equations through addition generates a new equation that reflects the combined relationships expressed by the original equations. For instance, adding the equations x + y = 5 and 2x – y = 4 yields 3x = 9. This process is fundamental in solving systems of equations, allowing for the elimination of variables and the determination of solutions that satisfy all involved equations.

This technique is crucial in diverse fields like physics, engineering, economics, and computer science. It provides a powerful tool for modeling complex systems, analyzing data, and making predictions. Historically, the development of methods for solving systems of equations marked a significant advancement in mathematics, enabling solutions to problems previously intractable. This ability to manipulate and combine equations laid the groundwork for many subsequent mathematical innovations.