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.

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

Short 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-by-step solution

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$D_x\mathrm{and}{\mathrm{D}}_{\mathrm{y}}$

localid="1647433211617" $$\begin{gathered} D_x=\left|\begin{array}{cc}0& -2\\ 4& 10\end{array}\right| \\ {D}_x=\left(0\right)\left(10\right)-\left(-2\right)\left(4\right) \\ {D}_x=0+8 \\ {D}_x=8 \end{gathered}$$

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 xand yare

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