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Answer :
Final Answer:
The vector form of the general solution for the given system of linear equations is [-2x₂ + x₃ - x₆, x₂, x₃, x₄, -x₆, x₆].
explanation:
In this system of linear equations, we have four equations with six variables: x₁, x₂, x₃, x₄, x₅, and x₆. To find the general solution, we first need to write the equations in matrix form (AX = 0), where A is the coefficient matrix and X is the column vector of variables.
Next, we perform row reduction on the augmented matrix [A|0] to bring it into its reduced row echelon form. This process allows us to express some variables in terms of others, leading to the identification of the basic and free variables.
After row reduction, we obtain the following equations:
x₂ = t₁
x₃ = t₁ + t₂
x₄ = -2t₁ - t₂
x₆ = t₂
Where t₁ and t₂ are arbitrary constants (parameters). These equations represent the solutions for the basic variables x₂, x₃, x₄, and x₆ in terms of the free variables t₁ and t₂.
Finally, we substitute the expressions for the basic variables back into the original equations, yielding the vector form of the general solution: [-2t₁ + t₂, t₁, t₁ + t₂, -2t₁ - t₂, t₂, t₂].
Linear systems can be solved using various methods such as Gaussian elimination, matrix inversion, and determinant properties. Understanding the concept of linear dependence and independence of equations is essential in solving systems of equations. It's also crucial to recognize the distinction between basic and free variables in order to express the general solution in vector form.
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