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.

$\left\{\begin{array}{l}2x-3y=-1\\ 10x+10y=5\end{array}\right.$

Short answer

The solution is$x=\frac{1}{10},\mathrm{y}=\frac{2}{5}\mathrm{and}\mathrm{in}\mathrm{ordered}\mathrm{pair}\mathrm{is}\left(\frac{1}{10},\frac{2}{5}\right).$

Step-by-step solution

Step 1. Given Information  

The given system of equation is

$\left\{\begin{array}{l}2x-3y=-1\\ 10x+10y=5\end{array}\right.$

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

$$\begin{gathered} D=\left|\begin{array}{cc}2& -3\\ 10& 10\end{array}\right| \\ D=\left(2\right)\left(10\right)-\left(-3\right)\left(10\right) \\ D=\left(20\right)-\left(-30\right) \\ D=50 \end{gathered}$$

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

Step 3. Use Cramer's rule 

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

$$\begin{gathered} D_x=\left|\begin{array}{cc}-1& -3\\ 5& 10\end{array}\right| \\ {D}_x=\left(-1\right)\left(10\right)-\left(-3\right)\left(5\right) \\ {D}_x=-10+15 \\ {D}_x=5 \end{gathered}$$

And

$$\begin{gathered} D_y=\left|\begin{array}{cc}2& -1\\ 10& 5\end{array}\right| \\ {D}_y=\left(2\right)\left(5\right)-\left(-1\right)\left(10\right) \\ {D}_y=10+10 \\ {D}_y=20 \end{gathered}$$

Step 4. Finding x and y  

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

$$\begin{gathered} x=\frac{D_x}{D}andy=\frac{D_y}{D} \\ x=\frac{5}{50}andy=\frac{20}{50} \\ x=\frac{1}{10}andy=\frac{2}{5} \end{gathered}$$