Looking for a common solution of two systems

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In summary: In particular, the vector z=x0 is a solution, but it is not a common solution to Ax=b and Cy=d. Therefore, the third system has more solutions than just the common solutions of the first two systems. In summary, given two systems Ax=b and Cy=d with at least one solution each, a common solution can be found by solving Az+Cz=b+d. However, this third system may have more solutions than just the common ones for the first and second systems.
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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.
 
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onako said:
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.
 

What is meant by "Looking for a common solution of two systems"?

"Looking for a common solution of two systems" refers to the process of finding a solution that satisfies both of the given systems of equations. This solution, if it exists, will satisfy both equations simultaneously.

What is the purpose of finding a common solution of two systems?

The purpose of finding a common solution of two systems is to determine the point of intersection between the two equations. This point represents the values of the variables that satisfy both equations and can provide valuable information in various applications.

How do you find the common solution of two systems?

To find the common solution of two systems, you can use various methods such as substitution, elimination, or graphing. These methods involve manipulating the equations to isolate a variable and then solving for its value. The resulting values can be substituted into the other equation to determine the common solution.

What does it mean if there is no common solution of two systems?

If there is no common solution of two systems, it means that the equations are parallel and do not intersect. This can occur when the equations have the same slope but different y-intercepts, or when the equations have different slopes. In either case, there is no point that satisfies both equations simultaneously.

Can there be more than one common solution of two systems?

Yes, it is possible to have more than one common solution of two systems. This occurs when the two equations are equivalent, meaning they represent the same line. In this case, every point on the line will satisfy both equations and can be considered a common solution.

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