Q 258 - Intermediate Algebra

Question

Use Cramer’s Rule to Solve Systems of Equations
In the following exercises, solve each system of equations using Cramer’s Rule.
$\left\{\begin{array}{l} 2x - 5y + 3z = 8 \\ 3x - y + 4z = 7\\ x + 3y + 2z = -3\end{array}\right.$

Step-by-Step Solution

Answer

The solution for the system of linear equation is,$\left(6,-1,-3\right)$.

Step 1 Given system of linear equation is,

$\left\{\begin{array}{l} 2x - 5y + 3z = 8 \\ 3x - y + 4z = 7\\ x + 3y + 2z = -3\end{array}\right.$

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

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

$$\begin{gathered} D=\left|\begin{array}{ccc}2& -5& 3\\ 3& -1& 4\\ 1& 3& 2\end{array}\right| \\ =2\left(-2-12\right)+5\left(6-4\right)+3\left(9+1\right) \\ =2\left(-14\right)+5\left(2\right)+3\left(10\right) \\ =-28+10+30 \\ =12 \end{gathered}$$

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

$$\begin{gathered} D_x=\left|\begin{array}{ccc}8& -5& 3\\ 7& -1& 4\\ -3& 3& 2\end{array}\right| \\ =8\left(-2-12\right)+5\left(14+12\right)+3\left(21-3\right) \\ =8\left(-14\right)+5\left(26\right)+3\left(18\right) \\ =-112+130+54 \\ =72 \end{gathered}$$

To find: $x$,

$$\begin{gathered} x=\frac{D_x}{D} \\ =\frac{72}{12} \\ =6 \end{gathered}$$

Step 4 Find the determinants D y by using the constants to replace the coefficients of y .

$$\begin{gathered} D_y=\left|\begin{array}{ccc}2& 8& 3\\ 3& 7& 4\\ 1& -3& 2\end{array}\right| \\ =2\left(14+12\right)-8\left(6-4\right)+3\left(-9-7\right) \\ =2\left(26\right)-8\left(2\right)+3\left(-16\right) \\ =52-16-48 \\ =-12 \end{gathered}$$

To find: $y$,

$$\begin{gathered} y=\frac{D_y}{D} \\ =\frac{-12}{12} \\ =-1 \end{gathered}$$

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

$$\begin{gathered} D_z=\left|\begin{array}{ccc}2& -5& 8\\ 3& -1& 7\\ 1& 3& -3\end{array}\right| \\ =2\left(3-21\right)+5\left(-9-7\right)+8\left(9+1\right) \\ =2\left(-18\right)+5\left(-16\right)+8\left(10\right) \\ =-36-80+80 \\ =-36 \end{gathered}$$

To find: $z$,

$$\begin{gathered} z=\frac{D_z}{D} \\ =\frac{-36}{12} \\ =-3 \end{gathered}$$

The solution for the system of linear equation is,$\left(6,-1,-3\right)$.

Step 6 write the solution as an ordered triad.

The ordered triad is, $\left(x,y,z\right)=\left(6,-1,-3\right)$.

Now substitute $x=6,y=-1,z=-3$ into the equation $2x-5y+3z=8$.

  $$\begin{gathered} 2x-5y+3z=8 \\ 2\left(6\right)-5\left(-1\right)+3\left(-3\right)=8 \\ 12+5-9=8 \\ 12-4=8 \\ 8=8 \end{gathered}$$

which is true.

Step 7 Substitute x = 6 , y = - 1 , z = - 3 into the equation 3 x - y + 4 z = 7 .

$$\begin{gathered} 3\left(6\right)-\left(-1\right)+4\left(-3\right)=7 \\ 18+1-12=7 \\ 7=7 \end{gathered}$$

which is true.