Finding Solutions for a System of Linear Equations with 2 Degrees of Freedom

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SUMMARY

The discussion focuses on solving a system of linear equations with two degrees of freedom, specifically analyzing the conditions under which the system has an infinite number of solutions. The system is represented in matrix form, and it is concluded that the value of k must be 2 for the system to exhibit these characteristics. The general solution derived includes parameters w, y, z, and x expressed in terms of t and s, confirming the correctness of the approach taken.

PREREQUISITES
  • Understanding of linear algebra concepts, particularly systems of linear equations.
  • Familiarity with elementary row operations in matrix manipulation.
  • Knowledge of degrees of freedom in the context of linear systems.
  • Ability to interpret and manipulate matrix representations of equations.
NEXT STEPS
  • Study the implications of degrees of freedom in linear systems.
  • Learn about the Rank-Nullity Theorem in linear algebra.
  • Explore parameterization of solutions in systems of equations.
  • Investigate the use of software tools like MATLAB or Python's NumPy for solving linear systems.
USEFUL FOR

Students and professionals in mathematics, particularly those studying linear algebra, as well as engineers and data scientists dealing with systems of equations in their work.

Yankel
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Hello again,

I have this system presented below:

\[\left\{\begin{matrix} x+y+z+w=1 \\ -3x-3y+kz-kw=-2 \\ 2x+2y+2z+kw=4-k \end{matrix}\right.\]

I need to find for which values of k the system has an infinite number of solutions with 2 degrees of freedom, and to find a general solution for this case. I did two elementary row operations to get this:

\[\begin{pmatrix} 1 &1 &1 &1 &1 \\ 0 &0 &k+3 &-k+3 &1 \\ 0 &0 &0 &k-2 &2-k \end{pmatrix}\]Then I said that infinite number of solutions with 2 DF will be when k=2, and my final solution was:

w=t
y=s
z=(1-t)/5
x=1-s-t-((1-t)/5)

Am I correct ?

Thanks !
 
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