Analyzing a System of Linear Equations

The question asks about the nature of the solutions for the given system of three linear equations in three variables: \(x\), \(y\), and \(z\).

• Equation 1: \(2x + y - 3z = 5\)
• Equation 2: \(3x - 2y + 2z = 5\)
• Equation 3: \(5x - 3y - z = 16\)

To determine if a system of linear equations is consistent (has at least one solution) or inconsistent (has no solution), and if consistent, whether it has a unique solution or infinitely many solutions, we can use methods like Gaussian elimination or determinants.

Understanding System Consistency and Solutions

A system of linear equations can fall into one of three categories:

• Consistent with a unique solution: There is exactly one set of values for the variables that satisfies all equations simultaneously.
• Consistent with infinitely many solutions: There are countless sets of values for the variables that satisfy all equations simultaneously. This happens when the equations are dependent.
• Inconsistent: There is no set of values for the variables that can satisfy all equations simultaneously. The equations contradict each other.

Solving the System using Determinants (Cramer's Rule)

For a system of \(n\) linear equations in \(n\) variables, we can use determinants to analyze the solutions. Consider the system \(Ax = B\), where \(A\) is the coefficient matrix, \(x\) is the variable vector, and \(B\) is the constant vector.

The coefficient matrix \(A\) for the given system is:

So, \(A = \begin{pmatrix} 2 & 1 & -3 \\ 3 & -2 & 2 \\ 5 & -3 & -1 \end{pmatrix}\).

The constant vector \(B\) is \(B = \begin{pmatrix} 5 \\ 5 \\ 16 \end{pmatrix}\).

Calculating the Determinant of the Coefficient Matrix

Let's calculate the determinant of the coefficient matrix \(A\), denoted as \(\Delta\) or \(\det(A)\).

\(\Delta = \det \begin{pmatrix} 2 & 1 & -3 \\ 3 & -2 & 2 \\ 5 & -3 & -1 \end{pmatrix}\)

Using cofactor expansion along the first row:

\(\Delta = 2 \times \det \begin{pmatrix} -2 & 2 \\ -3 & -1 \end{pmatrix} - 1 \times \det \begin{pmatrix} 3 & 2 \\ 5 & -1 \end{pmatrix} + (-3) \times \det \begin{pmatrix} 3 & -2 \\ 5 & -3 \end{pmatrix}\)

\(\Delta = 2 \times ((-2)(-1) - (2)(-3)) - 1 \times ((3)(-1) - (2)(5)) - 3 \times ((3)(-3) - (-2)(5))\)

\(\Delta = 2 \times (2 + 6) - 1 \times (-3 - 10) - 3 \times (-9 + 10)\)

\(\Delta = 2 \times (8) - 1 \times (-13) - 3 \times (1)\)

\(\Delta = 16 + 13 - 3\)

\(\Delta = 29 - 3\)

\(\Delta = 26\)

Interpreting the Determinant Value

The value of the determinant of the coefficient matrix, \(\Delta\), is 26.

For a system of linear equations \(Ax = B\):

• If \(\det(A) \neq 0\), the matrix \(A\) is invertible, and the system has a unique solution given by \(x = A^{-1}B\). This means the system is consistent with a unique solution.
• If \(\det(A) = 0\), the matrix \(A\) is not invertible. In this case, the system either has no solution (inconsistent) or infinitely many solutions (consistent with infinitely many solutions). Further analysis (e.g., checking the determinants \(\Delta_x, \Delta_y, \Delta_z\) in Cramer's rule or performing row reduction on the augmented matrix) is needed.

Since we calculated \(\Delta = 26\), which is not equal to zero, the system of equations is consistent and has a unique solution.

Conclusion on the System of Equations

Based on the determinant calculation, the given system of equations \(2x + y - 3z = 5\), \(3x - 2y + 2z = 5\), and \(5x - 3y - z = 16\) is consistent and has a unique solution.

Revision Table: System of Linear Equations

Additional Information: Methods for Solving Linear Systems

Besides determinants and Cramer's rule, other common methods to solve or analyze systems of linear equations include:

• Gaussian Elimination: Transforming the augmented matrix into row echelon form or reduced row echelon form using elementary row operations. This method directly reveals the nature of the solutions.
• Gauss-Jordan Elimination: A variation of Gaussian elimination that reduces the matrix further into reduced row echelon form, directly providing the unique solution if one exists.
• Matrix Inversion: If the coefficient matrix \(A\) is square and non-singular (\(\det(A) \neq 0\)), the unique solution is given by \(x = A^{-1}B\). This requires calculating the inverse of the matrix \(A\).

These methods provide alternative ways to confirm the nature of the solution for the system of equations.