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.
Get the complete, step-by-step math solution for: "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...". Powered by SolveForX AI math tutor.
Step-by-Step Solution
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∣=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 Ax, Ay, and Az. 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=∣A∣∣Ax∣, y=∣A∣∣Ay∣, and z=∣A∣∣Az∣. By substituting the calculated determinant values, we find the unique solution for the system.