Question

Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the solution set. See Examples 5–7. x + 2y + 3z = 4 4x + 3y + 2z = 1 -x - 2y - 3z = 0

Accepted Answer

Write the system of equations in matrix form as $$A\mathbf{x} = \mathbf{b}$$, where $$A is $$the coefficient matrix, $$\mathbf{x} is $$the column vector of variables, and $$\mathbf{b} is $$the constants vector. For this system, $$A = \begin{bmatrix} 1 & 2 & 3 \ 4 & 3 & 2 \ -1 & -2 & -3 \end{bmatrix}$$ and $$\mathbf{b} = \begin{bmatrix} 4 \ 1 \ 0 \end{bmatrix}. $$Calculate the determinant of the coefficient matrix $$D = \det(A). $$This will tell us if Cramer's rule can be applied. If $$D 
eq 0$$, the system has a unique solution; if $$D = 0$$, we need to use another method. If $$D 
eq 0$$, find the determinants $$D_x$$, $$D_y$$, and $$D_z by $$replacing the respective columns of $$A$$ with the vector $$\mathbf{b}. $$Specifically, $$D_x is $$the determinant of the matrix formed by replacing the first column of $$A$$ with $$\mathbf{b}$$, $$D_y$$ replaces the second column, and $$D_z$$ replaces the third column. Use Cramer's rule formulas to find the variables: $$x = \frac{D_x}{D}$$, $$y = \frac{D_y}{D}$$, and $$z = \frac{D_z}{D}. $$This gives the unique solution to the system. If $$D = 0$$, the system may have infinitely many solutions or no solution. In that case, use another method such as substitution or elimination to analyze the system further and determine the solution set.

Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the solution set. See Examples 5–7.
x + 2y + 3z = 4
4x + 3y + 2z = 1
-x - 2y - 3z = 0

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Key Concepts

Cramer's Rule

Determinant and Its Role in Systems of Equations

Alternative Methods for Solving Systems When D = 0