Question

Use Cramer’s Rule to Solve Systems of Equations.

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

$\left\{\begin{array}{l}2x+3y+z=12\\ x+y+z=9\\ 3x+4y+2z=20\end{array}\right.$

Short answer

The system is inconsistent and it has no solution.

Step-by-step solution

Given system of equations are,

$\left\{\begin{array}{l}2x+3y+z=12\\ x+y+z=9\\ 3x+4y+2z=20\end{array}\right.$

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.

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

We cannot use the 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, use the constants to replace the coefficients of x.

$$\begin{gathered} D_x=\left|\begin{array}{ccc}12& 3& 1\\ 9& 1& 1\\ 20& 4& 2\end{array}\right| \\ =12\left(2-4\right)-3\left(18-20\right)+1\left(36-20\right) \\ =12\left(-2\right)-3\left(-2\right)+1\left(16\right) \\ =-24+6+16 \\ =-2 \end{gathered}$$

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

$$\begin{gathered} D_y=\left|\begin{array}{ccc}2& 12& 1\\ 1& 9& 1\\ 3& 20& 2\end{array}\right| \\ =2\left(18-20\right)-12\left(2-3\right)+1\left(20-27\right) \\ =2\left(-2\right)-12\left(-1\right)+1\left(-7\right) \\ =-4+12-7 \\ =1 \end{gathered}$$

Step 6 Evaluating the determinant Dz, by using the constants to replace the coefficients of z.

$$\begin{gathered} D_z=\left|\begin{array}{ccc}2& 3& 12\\ 1& 1& 9\\ 3& 4& 20\end{array}\right| \\ =2\left(20-36\right)-3\left(20-27\right)+12\left(4-3\right) \\ =2\left(-16\right)-3\left(-7\right)+12\left(1\right) \\ =-32+21+12 \\ =1 \end{gathered}$$

All the determinants are not equal to zero.

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