Question

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

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

Short answer

The solution for the system of linear equation is,$\left(-3,-5,4\right)$.

Step-by-step solution

Given system of linear equation is,

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

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

Step 2 Find the determinants D by using the coefficients of the variable.

$$\begin{gathered} D=\left|\begin{array}{ccc}4& -3& 1\\ 2& -5& -4\\ 3& -2& -2\end{array}\right| \\ =4\left(10-8\right)+3\left(-4+12\right)+1\left(-4+15\right) \\ =4\left(2\right)+3\left(8\right)+1\left(11\right) \\ =8+24+11 \\ =43 \end{gathered}$$

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

$$\begin{gathered} D_x=\left|\begin{array}{ccc}7& -3& 1\\ 3& -5& -4\\ -7& -2& -2\end{array}\right| \\ =7\left(10-8\right)+3\left(-6-28\right)+1\left(-6-35\right) \\ =7\left(2\right)+3\left(-34\right)+1\left(-41\right) \\ =14-102-41 \\ =-129 \end{gathered}$$

To find: $x$,

$$\begin{gathered} x=\frac{D_x}{D} \\ =\frac{-129}{43} \\ =-3 \end{gathered}$$

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

$$\begin{gathered} D_y=\left|\begin{array}{ccc}4& 7& 1\\ 2& 3& -4\\ 3& -7& -2\end{array}\right| \\ =4\left(-6-28\right)-7\left(-4+12\right)+1\left(-14-9\right) \\ =4\left(-34\right)-7\left(8\right)+1\left(-23\right) \\ =-136-56-23 \\ =-215 \end{gathered}$$

To find: $y$,

$$\begin{gathered} y=\frac{D_y}{D} \\ =\frac{-215}{43} \\ =-5 \end{gathered}$$

Step 5 Find the determinant Dz by using the constants to replace the coefficients of z.

$$\begin{gathered} D_z=\left|\begin{array}{ccc}4& -3& 7\\ 2& -5& 3\\ 3& -2& -7\end{array}\right| \\ =4\left(35+6\right)+3\left(-14-9\right)+7\left(-4+15\right) \\ =4\left(41\right)+3\left(-23\right)+7\left(11\right) \\ =164-69+77 \\ =172 \end{gathered}$$

To find: $z$,

$$\begin{gathered} z=\frac{D_z}{D} \\ =\frac{172}{43} \\ =4 \end{gathered}$$

The solution for the system of linear equation is,$\left(-3,-5,4\right)$.

Step 6 write the solution as an ordered triad.

The ordered triad is, $\left(x,y,z\right)=\left(-3,-5,4\right)$.

Substitute $x=-3,y=-5,z=4$into the equation $4x-3y+z=7$.

$$\begin{gathered} 4\left(-3\right)-3\left(-5\right)+4=7 \\ -12+15+4=7 \\ 3+4=7 \\ 7=7 \end{gathered}$$

which is true.

Step 7 Now substitute x=-3,y=-5,z=4 into the equation 2x-5y-4z=3.

$$\begin{gathered} 2\left(-3\right)-5\left(-5\right)-4\left(4\right)=3 \\ -6+25-16=3 \\ 25-22=3 \\ 3=3 \end{gathered}$$

which is true.