Chapter 16.9, Problem 11OE

To determine

To calculate: The entries of the determinant $D_x$ for the system of equations,

  $\begin{array}{c}2x-3z=0\\ 3x+y=15\\ 4y+z=-8\end{array}$

Answer to Problem 11OE

The column one of determinant $D_x$ is $\left[\begin{array}{c}0\\ 15\\ -8\end{array}\right]$ , column two is $\left[\begin{array}{c}0\\ 1\\ 4\end{array}\right]$ and column three is $\left[\begin{array}{c}-3\\ 0\\ 1\end{array}\right]$ .

Explanation of Solution

Given information:

The system of equations are

  $\begin{array}{c}2x-3z=0\\ 3x+y=15\\ 4y+z=-8\end{array}$

Formula used:

For system of equations

  $\begin{array}{c}{a}_{1}x+b_{1}y+c_{1}z=d_1\\ a_{2}x+b_{2}y+c_{2}z=d_2\\ a_{3}x+b_{3}y+c_{3}z=d_3\end{array}$

The determinant $D_x$ is the coefficients of the variables y and z along with constant term in column form,

  $\left|\begin{array}{ccc}{d}_1& b_1& c_1\\ d_2& b_2& c_2\\ d_3& b_3& c_3\end{array}\right|$

Calculation:

Consider the system of equations are

  $\begin{array}{c}2x-3z=0\\ 3x+y=15\\ 4y+z=-8\end{array}$

Recall that for system of equations

  $\begin{array}{c}{a}_{1}x+b_{1}y+c_{1}z=d_1\\ a_{2}x+b_{2}y+c_{2}z=d_2\\ a_{3}x+b_{3}y+c_{3}z=d_3\end{array}$

The determinant $D_x$ is the coefficients of the variables y and z along with constant term in column form,

  $\left|\begin{array}{ccc}{d}_1& b_1& c_1\\ d_2& b_2& c_2\\ d_3& b_3& c_3\end{array}\right|$

Column one of determinant has constant term of all three equations.

Column two of determinant has coefficients of y in all three equations.

Column three of determinant has coefficients of z in all three equations.

  $D_x=\left|\begin{array}{ccc}0& 0& -3\\ 15& 1& 0\\ -8& 4& 1\end{array}\right|$

Thus, the column one of determinant $D_x$ is $\left[\begin{array}{c}0\\ 15\\ -8\end{array}\right]$ , column two is $\left[\begin{array}{c}0\\ 1\\ 4\end{array}\right]$ and column three is $\left[\begin{array}{c}-3\\ 0\\ 1\end{array}\right]$ .