Question

Use Cramer’s Rule to Solve Systems of Equations

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

$\left\{\begin{array}{l}-3x-y=4\\ 6x+2y=-16\end{array}\right.$

Short answer

The system is inconsistent and it has no solution.

Step-by-step solution

Given system of linear equations are,

$\left\{\begin{array}{l}-3x-y=4\\ 6x+2y=-16\end{array}\right.$

The objective is , we need to solve the given system of equations by using the Cramer's rule.

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

$$\begin{gathered} D=\left|\begin{array}{cc}-3& -1\\ 6& 2\end{array}\right| \\ =-3\left(2\right)-\left(-1\right)\left(6\right) \\ =-6+6 \\ =0 \end{gathered}$$

We cannot use Cramer's rule to solve this system. But by looking at the determinants $D_x,D_y$, we can determine whether the system is dependent or inconsistent.

Step 3Find the determinant Dxby using the constants to replace the coefficients of x.

$$\begin{gathered} D_x=\left|\begin{array}{cc}4& -1\\ -16& 2\end{array}\right| \\ =4\left(2\right)-\left(-1\right)\left(-16\right) \\ =8-16 \\ =-8 \end{gathered}$$

Step 4Find the determinant Dy by using the constants to replace the coefficients of y.

$$\begin{gathered} D_y=\left|\begin{array}{cc}-3& 4\\ 6& -16\end{array}\right| \\ =\left(-3\right)\left(-16\right)-4\left(6\right) \\ =48-24 \\ =24 \end{gathered}$$

Here, all the determinants are not equal to zero.

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