Question

Solve the system of equations using Cramer’s rule:

$\left\{\begin{array}{l}4x-3y=8\\ 8x-3y=14\end{array}\right.$

Short answer

The system is inconsistent. So, There is no solution.

Step-by-step solution

Step 1. Given information

The given linear equations are

$\left\{\begin{array}{l}4x-3y=8\\ 8x-3y=14\end{array}\right.$

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

$$\begin{gathered} D=\left[\begin{array}{cc}4& -3\\ 8& -6\end{array}\right] \\ D=(4\times -6)-(-3\times 8) \\ D=-24+24 \\ D=0 \end{gathered}$$

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

Step 3. Evaluate the determinant Dx and Dy. 

$$\begin{gathered} D_x=\left[\begin{array}{cc}8& -3\\ 14& -6\end{array}\right] \\ {D}_x=(8\times -6)-(-3\times 14) \\ {D}_x=-48+42 \\ {D}_x=-6 \\ {D}_y=\left[\begin{array}{cc}4& 8\\ 8& 14\end{array}\right] \\ {D}_y=(4\times 14)-(8\times 8) \\ {D}_y=56-64 \\ {D}_y=-8 \\ \mathrm{Since}\mathrm{all}\mathrm{the}\mathrm{determinants}\mathrm{are}\mathrm{not}\mathrm{zero},\mathrm{the}\mathrm{system}\mathrm{is}\mathrm{inconsistent}.\mathrm{There}\mathrm{is}\mathrm{no}\mathrm{solution}. \end{gathered}$$