Using Cramer's rule, show that the following system of linear equations is consistent and hence solve it : $x + y + z = 1$, $2x + 3y + 2z = 2$, $x + y + 2z = 4$.

Step 1: Form the coefficient matrix and constant matrix

First, we represent the given system of linear equations in matrix form $AX = B$, where $A$ is the coefficient matrix, $X$ is the variable matrix, and $B$ is the constant matrix. This setup is essential for applying Cramer's Rule.

Step 2: Calculate the determinant of the coefficient matrix

Next, we calculate the determinant of the coefficient matrix, denoted as $|A|$. If $|A| \neq 0$, the system is consistent and has a unique solution, allowing us to use Cramer's Rule. Here, the determinant is 1, which is non-zero.

Step 3: Calculate determinants for x, y, and z

To find the values of $x$, $y$, and $z$, we replace the respective column in the coefficient matrix $A$ with the constant matrix $B$ to form $A_x$, $A_y$, and $A_z$. We then calculate the determinants of these new matrices.

Step 4: Solve for x, y, and z using Cramer's Rule

Finally, we apply Cramer's Rule, which states that $x = \frac{|A_x|}{|A|}$, $y = \frac{|A_y|}{|A|}$, and $z = \frac{|A_z|}{|A|}$. By substituting the calculated determinant values, we find the unique solution for the system.

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