Question

In Problems 15 – 42, solve each system of equations using Cramer’s Rule if it is applicable. If Cramer’s Rule is not applicable, say so.

$\left\{\begin{array}{l}5x-y=13\\ 2x+3y=12\end{array}\right.$

Short answer

The solution is$x=3,\mathrm{y}=2\mathrm{and}\mathrm{in}\mathrm{ordered}\mathrm{pair}\mathrm{is}\left(3,2\right).$

Step-by-step solution

Step 1. Given Information 

The given system of equation is

$\left\{\begin{array}{l}5x-y=13\\ 2x+3y=12\end{array}\right.$

We have to solve the system of equations using Cramer's rule if it is applicable.

Step 2. Finding the determinant 

The determinantD of the coefficients of the variables is

$$\begin{gathered} D=\left|\begin{array}{cc}5& -1\\ 2& 3\end{array}\right| \\ D=\left(5\right)\left(3\right)-\left(-1\right)\left(2\right) \\ D=\left(15\right)-\left(-2\right) \\ D=17 \end{gathered}$$

Since$D\ne 0$ Cramer's rule is applicable.

Step 3. Use Cramer's rule 

Let's find$D_x\mathrm{and}{D}_y$

localid="1647428335291" $$\begin{gathered} D_x=\left|\begin{array}{cc}13& -1\\ 12& 3\end{array}\right| \\ {D}_x=\left(13\right)\left(3\right)-\left(-1\right)\left(12\right) \\ {D}_x=39+12 \\ {D}_x=51 \end{gathered}$$

And

localid="1647428350891" $$\begin{gathered} D_y=\left|\begin{array}{cc}5& 13\\ 2& 12\end{array}\right| \\ {D}_y=\left(5\right)\left(12\right)-\left(13\right)\left(2\right) \\ {D}_y=60-26 \\ {D}_y=34 \end{gathered}$$

Step 4. Finding x and y 

By using Cramer's rule, the values of xand yare

$$\begin{gathered} x=\frac{D_x}{D}andy=\frac{D_y}{D} \\ x=\frac{51}{17}andy=\frac{34}{17} \\ x=3andy=2 \end{gathered}$$