120k views
1 vote
If a system of linear equations t AX=B has infinitely many solutions, how many solutions will the system AX=C can have for some other C ? Give examples

1 Answer

2 votes

Final answer:

If a system of linear equations t AX=B has infinitely many solutions, it means the equations are not independent. The system AX=C can have zero, one, or infinitely many solutions, depending on the relationship of C to B and the original line/plane/space described by the equations.

Step-by-step explanation:

If a system of linear equations t AX=B has infinitely many solutions, this typically means that there is a dependency between the equations in the system, resulting in a situation where the equations are not independent. This usually happens when one equation is a multiple of the other, or both of them together describe the same line in the case of two variables, or plane/space in cases of three or more variables.

For the system AX=C with some other matrix C, it can have zero, one, or infinitely many solutions depending on how C relates to the original equations represented by B. If C falls on the same line/plane/space that B does, then the system AX=C will also have infinitely many solutions. If C does not align with that line/plane/space, the system will have no solutions because it represents a line/plane/space that does not intersect with the original. If the system of equations changes such that it becomes consistent and independent, then there will be a single solution.

Here's an example:

  • Original system: x + y = 2 (equation A) and 2x + 2y = 4 (equation B which is just 2*A).
  • Updated system: x + y = 3 (equation C).
  • The original system has infinitely many solutions because equations A and B are dependent. Equation C, however, does not have the same solutions as A and B, so the new system (A and C together) either has no solution if the lines represented do not intersect, or exactly one solution if they do.

User Keynslug
by
7.5k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.