Looking for a common solution of two systems

[onako]

Given two systems, Ax=b and Cy=d, for nxn matrices A and C, and n-dimensional vectors b and d, each of which has at least one solution, it is know that one solution is common to both (satisfies both equations). Could such solution be z found by solving Az+Cz=b+d? I understand that a common solution would satisfy the third system, but I wonder if the third system has more solutions, and we obtain one that is not common for the first and the second system.

 

[Erland]

Given two systems, Ax=b and Cy=d, for nxn matrices A and C, and n-dimensional vectors b and d, each of which has at least one solution, it is know that one solution is common to both (satisfies both equations). Could such solution be z found by solving Az+Cz=b+d? I understand that a common solution would satisfy the third system, but I wonder if the third system has more solutions, and we obtain one that is not common for the first and the second system.

It certainly can!

As an extreme example, let C=-A and d=-b, and assume that A is invertible. Then Ax=b has a unique solution, call it x₀. This x₀ is also a unique solution to By=d, so the systems have a unique common solution.

But A+C is a zero matrix, and b+d is a zero vector, so all vectors z in Rⁿ are solutions to (A+C)z=b+d.