Question

Use Cramer’s Rule to Solve Systems of Equations

In the following exercises, solve each system of equations using Cramer’s Rule.

$\left\{\begin{array}{l}4x+3y=2\\ 20x+15y=5\end{array}\right.$

Short answer

The system is inconsistent and it has no solution.

Step-by-step solution

Given system of equations are,

$\left\{\begin{array}{l}4x+3y=2\\ 20x+15y=5\end{array}\right.$

The objective is, we need to solve the given system of equations by using the Cramer's rule.

Step 2 Find the determinant D by using the coefficients of the variables.

$$\begin{gathered} D=\left|\begin{array}{cc}4& 3\\ 20& 15\end{array}\right| \\ =60-60 \\ =0 \end{gathered}$$

Here we cannot use Cramer's rule to solve this system. But by looking at the value of the determinants $D_x,D_y$, we can determine whether the system is dependent or inconsistent.

Step 3 Evaluate the determinant Dx by using the constants to replace the coefficients of x.

$$\begin{gathered} D_x=\left|\begin{array}{cc}2& 3\\ 5& 15\end{array}\right| \\ =2\left(15\right)-3\left(5\right) \\ =30-15 \\ =15 \end{gathered}$$

Step 4 Evaluate the determinant Dy by using the constants to replace the coefficients of y.

$$\begin{gathered} D_y=\left|\begin{array}{cc}4& 2\\ 20& 5\end{array}\right| \\ =4\left(5\right)-2\left(20\right) \\ =20-40 \\ =-20 \end{gathered}$$

Here all the determinants are not equal to zero.

So, the system is inconsistent and it has no solution.