Minimization solution of three equations in two variables

[barryj]

Homework Statement

given these three equations: I know I have more equations than variables. However, isn't there a way to find the closest solution, some sort of regression solution?
2x - y = -3
-2x - y = -4
-2.1x - y = 4

Relevant Equations

2x - y = -3
-2x - y = -4
-2.1x - y = 4

I do not know the solution.

 

[barryj]

I am thinking about how regression is performed. Let's assume I plot the three equations and they form a triange where they intersect. I can use the "distance from a point to a line" formula to get the distance from an arbitrary point , (X0,Y0) within the triangle to each of the lines. I think i could then square and add the distances and try to minimize the resulting function D(X0,Y0) . It seems that this could be extended to find the best fit of multiple lines or even planes. I guess this is the topic of optimization. I do not know if this is a good way or not.

 

[haruspex]

isn't there a way to find the closest solution, some sort of regression solution?

Given $AX=Y$, $A^TAX=A^TY$.
IF $A^TA$ is nonsingular you have $X=(A^TA)^{-1}A^TY$.
This can be shown to be the least sum squares solution.

 

[barryj]

Amazing! Hard to imagine it is this simple
Thanks.