Q. 17 - Precalculus Enhanced with Graphing Utilities

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. 

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

Step-by-Step Solution

Answer

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

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 determinant D 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 $D_x \mathrm{and} D_y$

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

And

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

Step 4. Finding x and y

By using Cramer's rule, the values of x and y are 

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