Analyzing the Given System of Linear Equations

We are given a system of three linear equations with three variables (x, y, and z):

• Equation 1: \(x + 2y + z = 6\)
• Equation 2: \(2x + y + 2z = 6\)
• Equation 3: \(x + y + z = 5\)

To determine the nature of the solution (unique solution, no solution, or infinite solutions) for this system of simultaneous equations, we can use the determinant method, also known as Cramer's rule principles.

First, we write the coefficient matrix A and the constant vector B:

Calculating the Determinant of the Coefficient Matrix

Next, we calculate the determinant of the coefficient matrix A, denoted as $\Delta$ or det(A).

Using the cofactor expansion along the first row:

Since $\Delta = 0$, the system does not have a unique solution. It will either have no solution or infinite solutions. To distinguish between these two cases, we calculate the determinants $\Delta_x$, $\Delta_y$, and $\Delta_z$, which are obtained by replacing the x, y, and z coefficient columns with the constant vector B, respectively.

Calculating Determinants for Variables

Calculate $\Delta_x$:

Using cofactor expansion along the first row:

Since $\Delta = 0$ and $\Delta_x \neq 0$, the system of equations is inconsistent. An inconsistent system has no solution.

We can also calculate $\Delta_y$ and $\Delta_z$ for completeness, but knowing $\Delta=0$ and at least one of $\Delta_x, \Delta_y, \Delta_z$ is non-zero is sufficient to conclude there is no solution.

Calculate $\Delta_y$:

Using cofactor expansion along the first row:

Calculate $\Delta_z$:

Using cofactor expansion along the first row:

Conclusion on System Solution

We found that:

• $\Delta = 0$
• $\Delta_x = 3 \neq 0$
• $\Delta_y = 0$
• $\Delta_z = -3 \neq 0$

According to the properties of determinants applied to a system of linear equations:

• If $\Delta \neq 0$, there is a unique solution.
• If $\Delta = 0$ and at least one of $\Delta_x, \Delta_y, \Delta_z$ is non-zero, there is no solution (inconsistent system).
• If $\Delta = 0$ and $\Delta_x = \Delta_y = \Delta_z = 0$, there are infinite solutions or no solution (further analysis like rank might be needed for the latter, but for typical systems derived from classroom problems, $\Delta_i=0$ when $\Delta=0$ usually implies infinite solutions).

In our case, $\Delta = 0$ and $\Delta_x \neq 0$ (and $\Delta_z \neq 0$), which means the system of simultaneous equation is inconsistent and has no solution.