# Solving a system of equations with numerical approximation [closed]

Let a, b, c be vectors obtained with the function N and x, y be scalars. I have the equation:

Solve[a == x b + y c, {x, y}]


Since I have used N, I get an empty solution, but I know that a solution exists.

Is there any way to find some sort of approximation such that the equation is satisfied?

I don't know how to do it. but maybe a solution with an interval for x and y?

$$x \pm c1$$ and $$y \pm c2$$

This is a linear equation, and you are probably looking for a least squares solution. So,

LeastSquares[Transpose[{b, c}], a]


should work.

To problem here is that there are either no or infinitely many (a line of) solutions.

By using Solve, you are asking Mathematica to solve for $$x$$ and $$y$$, which is not possible, as the system of equations is under defined. You can either solve for one of the unkowns

Solve[a == x b + y c, {x}]


or reduce the equations into simpler conditions

Reduce[a == x b + y c, {x, y}]