Question

Solve the system of equations using Cramer’s Rule if it is applicable. If Cramer’s Rule is not applicable, say so.

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

Short answer

$$\begin{gathered} x=6 \\ y=2 \end{gathered}$$

Step-by-step solution

Step 1. Given information

We have been given a system of equations:

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

We have to solve this system of equations using Cramer’s Rule if it is applicable.

Step 2. Check determinant 

$$\begin{gathered} \left|\begin{array}{cc}1& 1\\ 1& -1\end{array}\right| \\ =1\times (-1)-1\times 1 \\ =-1-1 \\ =-2 \end{gathered}$$

Since the determinant is non zero, so Cramer's rule is applicable.

Step 3. Use Cramer's rule

Find $D_x$and $D_y$.

$$\begin{gathered} D_x=\left|\begin{array}{cc}8& 1\\ 4& -1\end{array}\right| \\ =8\times (-1)-4\times 1 \\ =-8-4 \\ =-12 \end{gathered}$$

$$\begin{gathered} D_y=\left|\begin{array}{cc}1& 8\\ 1& 4\end{array}\right| \\ =1\times 4-1\times 8 \\ =4-8 \\ =-4 \end{gathered}$$

Step 4. Find x and y

$$\begin{gathered} x=\frac{D_x}{D} \\ =\frac{-12}{-2} \\ =6 \\ y=\frac{D_y}{D} \\ =\frac{-4}{-2} \\ =2 \end{gathered}$$