Q. 21 - 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. 

$\left\{\begin{array}{l}3x-6y=24\\ 5x+4y=12\end{array}\right.$

Step-by-Step Solution

Answer

The solution is $x=4, \mathrm{y}=-2 \mathrm{and} \mathrm{in} \mathrm{ordered} \mathrm{pair} \mathrm{is} \left(4,-2\right).$

Step 1. Given Information

The given system of equation is

$\left\{\begin{array}{l}3x-6y=24\\ 5x+4y=12\end{array}\right.$

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 

$$\begin{gathered} D=\left|\begin{array}{cc}3& -6\\ 5& 4\end{array}\right| \\ D=\left(3\right)\left(4\right)-\left(-6\right)\left(5\right) \\ D=12+30 \\ D=42 \end{gathered}$$

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

Step 3. Use Cramer's rule

Let's find $D_x \mathrm{and} D_y$

$$\begin{gathered} D_x=\left|\begin{array}{cc}24& -6\\ 12& 4\end{array}\right| \\ {D}_x=\left(24\right)\left(4\right)-\left(-6\right)\left(12\right) \\ {D}_x=96+72 \\ {D}_x=168 \end{gathered}$$

And

$$\begin{gathered} D_y=\left|\begin{array}{cc}3& 24\\ 5& 12\end{array}\right| \\ {D}_y=\left(3\right)\left(12\right)-\left(24\right)\left(5\right) \\ {D}_y=36-120 \\ {D}_y=-84 \end{gathered}$$

Step 4. Finding x and y

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

$$\begin{gathered} x=\frac{D_x}{D} and y=\frac{D_y}{D} \\ x=\frac{168}{42} and y=\frac{-84}{42} \\ x=4 and y=-2 \end{gathered}$$