Question

Use Cramer’s Rule to Solve Systems of Equations

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

$\left\{\begin{array}{l}2x+y=3\\ 6x+3y=9\end{array}\right.$

Short answer

The system is consistent and dependent and having infinitely many solutions.

Step-by-step solution

Given system of linear equations are,

$\left\{\begin{array}{l}2x+y=3\\ 6x+3y=9\end{array}\right.$

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

Step 2 Find the determinant D by using the coefficients of the variables.

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

Here we cannot solve this system by using the Cramer's rule. But by looking at the determinant $D_x,D_y$, we can determine whether the system is dependent or inconsistent.

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

$$\begin{gathered} D_x=\left|\begin{array}{cc}3& 1\\ 9& 3\end{array}\right| \\ =3\left(3\right)-1\left(9\right) \\ =9-9 \\ =0 \end{gathered}$$

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

$$\begin{gathered} D_y=\left|\begin{array}{cc}2& 3\\ 6& 9\end{array}\right| \\ =2\left(9\right)-3\left(6\right) \\ =18-18 \\ =0 \end{gathered}$$

Here all the determinants are zero.

So, the system is consistent and dependent and have infinitely many solutions.