Question

Solve the system of equations using Cramer’s rule:$\left\{\begin{array}{l}x+3y=4\\ 2x+6y=8\end{array}\right.$

Short answer

Here, all the determinants are equal to zero.

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

Step-by-step solution

Step 1. Given information

The given linear equations are

$\left\{\begin{array}{l}x+3y=4\\ 2x+6y=8\end{array}\right.$

Step 2. Evaluate the determinant D, using the coefficients of he variables 

$$\begin{gathered} D=\left[\begin{array}{cc}1& 3\\ 2& 6\end{array}\right] \\ D=1\times 6-3\times 2 \\ D=6-6 \\ D=0 \end{gathered}$$

We cannot use Cramer’s Rule to solve this system. But by looking at the value of the determinants Dx and Dy, we can determine whether the system is dependent or inconsistent.

Step 3. Evaluate the determinant Dx.  

$$\begin{gathered} D_x=\left[\begin{array}{cc}4& 3\\ 8& 6\end{array}\right] \\ {D}_x=4\times 6-3\times 8 \\ {D}_x=24-24 \\ {D}_x=0 \\ \mathrm{Here},\mathrm{all}\mathrm{the}\mathrm{determinants}\mathrm{are}\mathrm{equal}\mathrm{to}\mathrm{zero}. \\ \mathrm{So},\mathrm{the}\mathrm{system}\mathrm{is}\mathrm{consistent}\mathrm{and}\mathrm{dependent}\mathrm{and}\mathrm{it}\mathrm{has}\mathrm{infinitely}\mathrm{many}\mathrm{solutions}. \end{gathered}$$