Equivalent systems of equations | How to transform a system into an equivalent one
Equivalent systems of equations | systems of equations |solving systems of equations |system of linear equations
Equivalent systems of equations:
Two Equivalent systems of equations are equivalent if and only if they have the same set of solutions. In other words, two systems are equivalent if and only if every solution of one of them is also a solution of the other.
All the main methods used to solve linear systems are based on the same principle: given a system, we transform it into an equivalent system that is easier to solve; then, its solution is also the solution of the original system.
Definition:
Remember that a system of equations in unknowns can be written in matrix form as
Ax=bwhere is the matrix of coefficients of the system, is the vector of constants and is the vector of unknowns.
Definition LetA1x=b1andA2x=b2be two linear systems, both having an vector of unknowns. The two systems are equivalent if and only if they have the same solutions, that is, if and only if![[eq4]](https://www.statlect.com/images/equivalent-systems-of-equations__14.png){kind=small}
How to transform a system into an equivalent one:
Proposition The system of equations in unknowns
is equivalent to the system for an invertible matrix.
Example:
Consider the system of two equations in two unknowns
The system can be written in matrix form as
where
![[eq19]](https://www.statlect.com/images/equivalent-systems-of-equations__43.png)
If we multiply the first equation by and leave the second equation unchanged, we obtain a new system
The matrix form of the new system is
where
![[eq22]](https://www.statlect.com/images/equivalent-systems-of-equations__47.png)
The new system is equivalent to the original one because the same result can be achieved by pre-multiplying the matrix form of the original system by the invertible matrixIn fact,
![[eq24]](https://www.statlect.com/images/equivalent-systems-of-equations__49.png)
and![[eq25]](https://www.statlect.com/images/equivalent-systems-of-equations__50.png)
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