Question

Use Cramer’s Rule to Solve Systems of Equations In the following exercises, solve each system of equations using Cramer’s Rule.

$\left\{\begin{array}{l}x-2y+3z=1\\ x+y-3z=7\\ 3x-4y+5z=7\end{array}\right.$

Short answer

The system is consistent and dependent and it has infinitely many solutions.

Step-by-step solution

Given system of equations are,

$\left\{\begin{array}{l}x-2y+3z=1\\ x+y-3z=7\\ 3x-4y+5z=7\end{array}\right.$

To find:

We need to find the solution for the given system of equations by using Cramer's rule.

Step 3 First find the determinant D by using the coefficients of the variables.

$$\begin{gathered} D=\left|\begin{array}{ccc}1& -2& 3\\ 1& 1& -3\\ 3& -4& 5\end{array}\right| \\ D=1\left(5-12\right)+2\left(5+9\right)+3\left(-4-3\right) \\ =1\left(-7\right)+2\left(14\right)+3\left(-7\right) \\ =-7+28-21 \\ =0 \end{gathered}$$

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

Step 4 Evaluate the determinant Dx, use the constants to replace the coefficients of x.

$$\begin{gathered} D_x=\left|\begin{array}{ccc}1& -2& 3\\ 7& 1& -3\\ 7& -4& 5\end{array}\right| \\ =1\left(5-12\right)+2\left(35+21\right)+3\left(-28-7\right) \\ =1\left(-7\right)+2\left(56\right)+3\left(-35\right) \\ =-7+112-105 \\ =0 \end{gathered}$$

Step 5 Now evaluate the determinant Dy, use the constants to replace the coefficients of y.

$$\begin{gathered} D_y=\left|\begin{array}{ccc}1& 1& 3\\ 1& 7& -3\\ 3& 7& 5\end{array}\right| \\ =1\left(35+21\right)-1\left(5+9\right)+3\left(7-21\right) \\ =1\left(56\right)-1\left(14\right)+3\left(-14\right) \\ =56-14-42 \\ =0 \end{gathered}$$

Step 6 Evaluate the determinant Dz, use the constants to replace the coefficients of z.

$$\begin{gathered} D_z=\left|\begin{array}{ccc}1& -2& 1\\ 1& 1& 7\\ 3& -4& 7\end{array}\right| \\ =1\left(7+28\right)+2\left(7-21\right)+1\left(-4-3\right) \\ =1\left(35\right)+2\left(-14\right)+1\left(-7\right) \\ =35-28-7 \\ =0 \end{gathered}$$

Here, all the determinants are equal to zero.

So, the system is consistent and dependent and it has infinitely many solutions.