Chapter 7.5, Problem 58E

(a).

To determine

To verify: the matrix is solution of the system of linear equation.

Answer to Problem 58E

The matrix $\left[\begin{array}{c}-1\\ 3\end{array}\right]$ is the solution of the given system of equation .

Explanation of Solution

Given information:

1. The system of linear equation is as follows.

  $\begin{array}{l}6x+2y=0\\ -x+5y=16\end{array}$

2. The matrix which needs to be verified as solution of system of linear equation is as follows

  $\left[\begin{array}{c}-1\\ 3\end{array}\right]$ ..

Calculation:

The system of linear equation is as follows.

  $\begin{array}{l}6x+2y=0\\ -x+5y=16\end{array}$

Representing system of linear equation in matrix form

  $\left[\begin{array}{cc}6& 2\\ -1& 5\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}0\\ 16\end{array}\right]\mathrm{...}(1)$

Substitute $\left[\begin{array}{c}x\\ y\end{array}\right]$ as $\left[\begin{array}{c}-1\\ 3\end{array}\right]$ in equation (1) and solve Left Hand Side to verify the solution

  $\left[\begin{array}{cc}6& 2\\ -1& 5\end{array}\right]\left[\begin{array}{c}-1\\ 3\end{array}\right]=\left[\begin{array}{c}6\times (-1)+2\times 3\\ (-1)\times (-1)+5\times 3\end{array}\right]$

  $=\left[\begin{array}{c}-6+6\\ 1+15\end{array}\right]$

  $=\left[\begin{array}{c}0\\ 16\end{array}\right]$

Since Left Hand Side of the equation is equal to Right Hand Side, the matrix $\left[\begin{array}{c}-1\\ 3\end{array}\right]$ is the solution of the given system of equation .

(b).

To determine

To verify: the matrix is solution of the system of linear equation.

Answer to Problem 58E

The matrix $\left[\begin{array}{c}2\\ -6\end{array}\right]$ is not the solution of the given system of equation .

Explanation of Solution

Given information:

1. The system of linear equation is as follows.

  $\begin{array}{l}6x+2y=0\\ -x+5y=16\end{array}$

2. The matrix which needs to be verified as solution of system of linear equation is as follows

  $\left[\begin{array}{c}2\\ -6\end{array}\right]$ ..

Calculation:

The system of linear equation is as follows.

  $\begin{array}{l}6x+2y=0\\ -x+5y=16\end{array}$

Representing system of linear equation in matrix form

  $\left[\begin{array}{cc}6& 2\\ -1& 5\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}0\\ 16\end{array}\right]\mathrm{...}(1)$

Substitute $\left[\begin{array}{c}x\\ y\end{array}\right]$ as $\left[\begin{array}{c}2\\ -6\end{array}\right]$ in equation (1) and solve Left Hand Side to verify the solution

  $\left[\begin{array}{cc}6& 2\\ -1& 5\end{array}\right]\left[\begin{array}{c}2\\ -6\end{array}\right]=\left[\begin{array}{c}6\times 2+2\times (-6)\\ (-1)\times 2+5\times (-6)\end{array}\right]$

  $=\left[\begin{array}{c}12-12\\ -2-30\end{array}\right]$

  $=\left[\begin{array}{c}0\\ -32\end{array}\right]$

Since Left Hand Side of the equation is not equal to Right Hand Side, the matrix $\left[\begin{array}{c}2\\ -6\end{array}\right]$ is not the solution of the given system of equation .

(c).

To determine

To verify: the matrix is solution of the system of linear equation.

Answer to Problem 58E

The matrix $\left[\begin{array}{c}3\\ -9\end{array}\right]$ is not the solution of the given system of equation .

Explanation of Solution

Given information:

1. The system of linear equation is as follows.

  $\begin{array}{l}6x+2y=0\\ -x+5y=16\end{array}$

2. The matrix which needs to be verified as solution of system of linear equation is as follows

  $\left[\begin{array}{c}3\\ -9\end{array}\right]$ ..

Calculation:

The system of linear equation is as follows.

  $\begin{array}{l}6x+2y=0\\ -x+5y=16\end{array}$

Representing system of linear equation in matrix form

  $\left[\begin{array}{cc}6& 2\\ -1& 5\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}0\\ 16\end{array}\right]\mathrm{...}(1)$

Substitute $\left[\begin{array}{c}x\\ y\end{array}\right]$ as $\left[\begin{array}{c}3\\ -9\end{array}\right]$ in equation (1) and solve Left Hand Side to verify the solution

  $\left[\begin{array}{cc}6& 2\\ -1& 5\end{array}\right]\left[\begin{array}{c}3\\ -9\end{array}\right]=\left[\begin{array}{c}6\times 3+2\times (-9)\\ (-1)\times 3+5\times (-9)\end{array}\right]$

  $=\left[\begin{array}{c}18-18\\ -3-45\end{array}\right]$

  $=\left[\begin{array}{c}0\\ -48\end{array}\right]$

Since Left Hand Side of the equation is not equal to Right Hand Side, the matrix $\left[\begin{array}{c}3\\ -9\end{array}\right]$ is not the solution of the given system of equation .

(d).

To determine

To verify: the matrix is solution of the system of linear equation.

Answer to Problem 58E

The matrix $\left[\begin{array}{c}-3\\ 9\end{array}\right]$ is not the solution of the given system of equation .

Explanation of Solution

Given information:

1. The system of linear equation is as follows.

  $\begin{array}{l}6x+2y=0\\ -x+5y=16\end{array}$

2. The matrix which needs to be verified as solution of system of linear equation is as follows

  $\left[\begin{array}{c}-3\\ 9\end{array}\right]$ ..

Calculation:

The system of linear equation is as follows.

  $\begin{array}{l}6x+2y=0\\ -x+5y=16\end{array}$

Representing system of linear equation in matrix form

  $\left[\begin{array}{cc}6& 2\\ -1& 5\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}0\\ 16\end{array}\right]\mathrm{...}(1)$

Substitute $\left[\begin{array}{c}x\\ y\end{array}\right]$ as $\left[\begin{array}{c}-3\\ 9\end{array}\right]$ in equation (1) and solve Left Hand Side to verify the solution

  $\left[\begin{array}{cc}6& 2\\ -1& 5\end{array}\right]\left[\begin{array}{c}-3\\ 9\end{array}\right]=\left[\begin{array}{c}6\times (-3)+2\times 9\\ (-1)\times (-3)+5\times 9\end{array}\right]$

  $=\left[\begin{array}{c}-18+18\\ 3+45\end{array}\right]$

  $=\left[\begin{array}{c}0\\ 48\end{array}\right]$

Since Left Hand Side of the equation is not equal to Right Hand Side, the matrix $\left[\begin{array}{c}-3\\ 9\end{array}\right]$ is not the solution of the given system of equation .