Question

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

$\left\{\begin{array}{l}11x+9y+2z=-9\\ 7x+5y+3z=-7\\ 4x+3y+z=-3\end{array}\right.$

Short answer

The solution for the system of linear equation is,$\left(2,-3,-2\right)$.

Step-by-step solution

Given system of linear equations is,

$\left\{\begin{array}{l}11x+9y+2z=-9\\ 7x+5y+3z=-7\\ 4x+3y+z=-3\end{array}\right.$

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

Find the determinant D by using the coefficients of the variable.

$$\begin{gathered} D=\left|\begin{array}{ccc}11& 9& 2\\ 7& 5& 3\\ 4& 3& 1\end{array}\right| \\ =11\left(5-9\right)-9\left(7-12\right)+2\left(21-20\right) \\ =11\left(-4\right)-9\left(-5\right)+2\left(1\right) \\ =-44+45+2 \\ =3 \end{gathered}$$

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

$$\begin{gathered} D_x=\left|\begin{array}{ccc}-9& 9& 2\\ -7& 5& 3\\ -3& 3& 1\end{array}\right| \\ =-9\left(5-9\right)-9\left(-7+9\right)+2\left(-21+15\right) \\ =-9\left(-4\right)-9\left(2\right)+2\left(-6\right) \\ =36-18-12 \\ =6 \end{gathered}$$

To find: $x$,

$$\begin{gathered} x=\frac{D_x}{D} \\ =\frac{6}{3} \\ x=2 \end{gathered}$$

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

$$\begin{gathered} D_y=\left|\begin{array}{ccc}11& -9& 2\\ 7& -7& 3\\ 4& -3& 1\end{array}\right| \\ =11\left(-7+9\right)+9\left(7-12\right)+2\left(-21+28\right) \\ =11\left(2\right)+9\left(-5\right)+2\left(7\right) \\ =22-45+14 \\ =-9 \end{gathered}$$

To find: $y$,

$$\begin{gathered} y=\frac{D_y}{D} \\ =\frac{-9}{3} \\ =-3 \end{gathered}$$

Step 5 Find the determinants Dz by using the constants to replace the coefficient of z.

$$\begin{gathered} D_z=\left|\begin{array}{ccc}11& 9& -9\\ 7& 5& -7\\ 4& 3& -3\end{array}\right| \\ =11\left(-15+21\right)-9\left(-21+28\right)-9\left(21-20\right) \\ =11\left(6\right)-9\left(7\right)-9\left(1\right) \\ =66-63-9 \\ =-6 \end{gathered}$$

To find: $z$,

$$\begin{gathered} z=\frac{D_z}{D} \\ =\frac{-6}{3} \\ =-2 \end{gathered}$$

The solution for the system of linear equation is,$\left(2,-3,-2\right)$.

Step 6 write the solution as an ordered triad.

The ordered triad is,$\left(x,y,z\right)=\left(2,-3,-2\right)$.

Now substitute $x=2,y=-3,z=-2$into the equation $7x+5y+3z=-7$.

$$\begin{gathered} 7\left(2\right)+5\left(-3\right)+3\left(-2\right)=-7 \\ 14-15-6=-7 \\ 14-21=-7 \\ -7=-7 \end{gathered}$$

which is true.

Step 7 Substitute x=2,y=-3,z=-2 into the equation 4x+3y+z=-3.

$$\begin{gathered} 4\left(2\right)+3\left(-3\right)+\left(-2\right)=-3 \\ 8-9-2=-3 \\ 8-11=-3 \\ -3=-3 \end{gathered}$$

which is true.