Given ##AX=Y##, ##A^TAX=A^TY##.barryj said:isn't there a way to find the closest solution, some sort of regression solution?
The minimization solution of three equations in two variables refers to finding the values of the two variables that simultaneously satisfy all three equations while minimizing the overall error or difference between the equations.
Finding the minimization solution allows us to accurately model and predict real-world phenomena by reducing the error between our equations and the actual data. It is also a fundamental problem in optimization and has numerous applications in fields such as economics, engineering, and physics.
The most commonly used methods include the substitution method, elimination method, and graphing method. Other more advanced techniques such as matrix operations, gradient descent, and calculus-based optimization can also be used.
One of the main challenges is that there is no guaranteed method to find the exact solution for all cases. Depending on the equations and initial values, some methods may not converge or may give inaccurate results. It also requires a good understanding of algebra and problem-solving skills.
The minimization solution can be used in various real-world scenarios such as finding the optimal production levels in a manufacturing process, determining the most profitable investment portfolio, or predicting the trajectory of a projectile. It can also be used in machine learning algorithms to minimize the error between predicted and actual values.