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.
