I have a linear equation: $a*x_1-b*x_2+x_3=1.0$. $a$ and $b$ are known parameters: $a=b=10e+6$. We used an approach to get the estimates of $x_1$ and $x_2$: $x_1=1.001$ and $x_2=0.9998$. We want to estimate $x_3$ Using this linear equation.
I intentionally created this problem so that I know the true values of $x1$ and $x2$. The true solution of this equation is: $x_1=x_2=x_3=1.0$. But because there are noise in the estimates of $x_1$ and $x_2$, we got a weird estimate of $x_3$: $x_3=-1199.0$, a very big value.
I am looking for a correct approach to obtain the estimate of $x_3$ Which should be around 1.0.