Q. 28 - 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. 

$\left\{\begin{array}{l}3x-2y=0\\ 5x+10y=4\end{array}\right.$

Step-by-Step Solution

Answer

The solution is $x=\frac{1}{5}, \mathrm{y}=\frac{3}{10} \mathrm{and} \mathrm{in} \mathrm{ordered} \mathrm{pair} \mathrm{is} \left(\frac{1}{5},\frac{3}{10}\right).$

Step 1. Given Information

The given system of equation is

$\left\{\begin{array}{l}3x-2y=0\\ 5x+10y=4\end{array}\right.$

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

$$\begin{gathered} D=\left|\begin{array}{cc}3& -2\\ 5& 10\end{array}\right| \\ D=\left(3\right)\left(10\right)-\left(-2\right)\left(5\right) \\ D=\left(30\right)-\left(-10\right) \\ D=40 \end{gathered}$$

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

Step 3. Use Cramer's rule

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

$$\begin{gathered} D_x=\left|\begin{array}{cc}0& -2\\ 4& 10\end{array}\right| \\ {D}_x=\left(0\right)\left(10\right)-\left(-2\right)\left(4\right) \\ {D}_x=0+8 \\ {D}_x=8 \end{gathered}$$

And

$$\begin{gathered} D_y=\left|\begin{array}{cc}3& 0\\ 5& 4\end{array}\right| \\ {D}_y=\left(3\right)\left(4\right)-\left(0\right)\left(5\right) \\ {D}_y=12-0 \\ {D}_y=12 \end{gathered}$$

Step 4. Finding x and y

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

$$\begin{gathered} x=\frac{D_x}{D} and y=\frac{D_y}{D} \\ x=\frac{8}{40} and y=\frac{12}{40} \\ x=\frac{1}{5} and y=\frac{3}{10} \end{gathered}$$