Question

Use Cramer’s Rule to Solve Systems of Equations

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

$\left\{\begin{array}{l}x-4y=-1\\ -3x+12y=3\end{array}\right.$

Short answer

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

Step-by-step solution

Given system of linear equations are,

$\left\{\begin{array}{l}x-4y=-1\\ -3x+12y=3\end{array}\right.$

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.

$$\begin{gathered} D=\left|\begin{array}{cc}1& -4\\ -3& 12\end{array}\right| \\ =1\left(12\right)-\left(-4\right)\left(-3\right) \\ =12-12 \\ =0 \end{gathered}$$

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

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

$$\begin{gathered} D_x=\left|\begin{array}{cc}-1& -4\\ 3& 12\end{array}\right| \\ =\left(-1\right)\left(12\right)-\left(-4\right)\left(3\right) \\ =-12+12 \\ =0 \end{gathered}$$

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

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

All the determinants are equal to zero.

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