Question

Use Cramer’s Rule to Solve Systems of Equations.

In the following exercises, solve each system of equations using Cramer’s Rule.

$\left\{\begin{array}{l}3x+4y-3z=-2\\ 2x+3y-z=-12\\ x+y-2z=6\end{array}\right.$

Short answer

The system is inconsistent and it has no solution.

Step-by-step solution

Given system of equations are,

$\left\{\begin{array}{l}3x+4y-3z=-2\\ 2x+3y-z=-12\\ x+y-2z=6\end{array}\right.$

to find:

We need to find the solution for the system of equations by using the Cramer's rule.

Step 3 Find the determinant D by using the coefficients of the variables.

$$\begin{gathered} D=\left|\begin{array}{ccc}3& 4& -3\\ 2& 3& -1\\ 1& 1& -2\end{array}\right| \\ =3\left(-6+1\right)-4\left(-4+1\right)-3\left(2-3\right) \\ =3\left(-5\right)-4\left(-3\right)-3\left(-1\right) \\ =-15+12+3 \\ =0 \end{gathered}$$

Here, we cannot use Cramer's rule to solve this system.

But by looking at the value of the determinants $D_x,D_y,D_z$, we can determine whether the system is dependent or inconsistent.

Step 4 Evaluate the determinant Dx by using the constants to replace the coefficients of x.

$$\begin{gathered} D_x=\left|\begin{array}{ccc}-2& 4& -3\\ -12& 3& -1\\ 6& 1& -2\end{array}\right| \\ =-2\left(-6+1\right)-4\left(24+6\right)-3\left(-12-18\right) \\ =-2\left(-5\right)-4\left(30\right)-3\left(-30\right) \\ =10-120+90 \\ =-20 \end{gathered}$$

Step 5 Evaluate the determinant Dy, by using the constants to replace the coefficients of y.

$$\begin{gathered} D_y=\left|\begin{array}{ccc}3& -2& -3\\ 2& -12& -1\\ 1& 6& -2\end{array}\right| \\ =3\left(24+6\right)+2\left(-4+1\right)-3\left(12+12\right) \\ =3\left(30\right)+2\left(-3\right)-3\left(24\right) \\ =90-6-72 \\ =12 \end{gathered}$$

Step 6 Evaluate the determinant Dz use the constants to replace the coefficients of z.

$$\begin{gathered} D_z=\left|\begin{array}{ccc}3& 4& -2\\ 2& 3& -12\\ 1& 1& 6\end{array}\right| \\ =3\left(18+12\right)-4\left(12+12\right)-2\left(2-3\right) \\ =3\left(30\right)-4\left(24\right)-2\left(-1\right) \\ =90-96+2 \\ =-4 \end{gathered}$$

All the determinants are not equal to zero.

So, the system is inconsistent and it has no solution.