Westonci.ca offers fast, accurate answers to your questions. Join our community and get the insights you need now. Get quick and reliable solutions to your questions from a community of experienced professionals on our platform. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
Let's analyze how to solve this system of equations using matrix multiplication. We are given the following system of equations:
[tex]\[ \begin{array}{l} x + y + z = 160 \\ x - 2y - z = -100 \\ 2x + 3y + 2z = 360 \\ \end{array} \][/tex]
This system can be expressed in matrix form as [tex]\( A \mathbf{x} = \mathbf{B} \)[/tex], where:
[tex]\[ A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & -2 & -1 \\ 2 & 3 & 2 \end{bmatrix}, \quad \mathbf{x} = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, \quad \text{and} \quad B = \begin{bmatrix} 160 \\ -100 \\ 360 \end{bmatrix} \][/tex]
Next, let's evaluate each option to find the one that represents the solution.
### Option 1:
[tex]\[ \left[\begin{array}{ccc}1 & 1 & 1 \\ 1 & -2 & -1 \\ 2 & 3 & 2\end{array}\right] \left[\begin{array}{c}160 \\ -100 \\ 360\end{array}\right] \][/tex]
Multiplying the matrix [tex]\( A \)[/tex] by vector [tex]\( B \)[/tex] gives [tex]\(A \mathbf{B}\)[/tex]. This operation uniquely resolves to check if [tex]\( A \mathbf{B} \)[/tex] results in [tex]\( B \)[/tex] which makes the system true. Based on the analysis:
[tex]\[ A \begin{bmatrix} 160 \\ -100 \\ 360 \end{bmatrix} = \begin{bmatrix} 160 \\ -100 \\ 360 \end{bmatrix} \][/tex]
Therefore, option 1 provides the correct matrix equation representing the solution.
### Option 2:
[tex]\[ \left[\begin{array}{ccc}0.5 & -0.5 & 0.5 \\ 2 & 0 & -1 \\ 3.5 & 0.5 & 1.5\end{array}\right] \left[\begin{array}{c}360 \\ -100 \\ 160\end{array}\right] \][/tex]
Evaluating this multiplication, it does not result in [tex]\( \mathbf{B} = \begin{bmatrix} 160 \\ -100 \\ 360 \end{bmatrix} \)[/tex].
### Option 3:
[tex]\[ \left[\begin{array}{ccc}-0.5 & 0.5 & 0.5 \\ -2 & 0 & 1 \\ 3.5 & -0.5 & -1.5\end{array}\right] \left[\begin{array}{c}160 \\ -100 \\ 360\end{array}\right] \][/tex]
Similarly, evaluating this multiplication does not result in [tex]\( \mathbf{B} \)[/tex].
### Option 4:
[tex]\[ \left[\begin{array}{ccc}-1 & -1 & -1 \\ -1 & 2 & 1 \\ -2 & -3 & -2\end{array}\right] \left[\begin{array}{c}360 \\ -100 \\ 160\end{array}\right] \][/tex]
Likewise, evaluating this multiplication, does not result in [tex]\( \mathbf{B} \)[/tex].
### Conclusion:
The option that satisfies [tex]\( A \mathbf{x} = \mathbf{B} \)[/tex] and correctly represents the solution to the system of equations is:
[tex]\[ \left[\begin{array}{ccc}1 & 1 & 1 \\ 1 & -2 & -1 \\ 2 & 3 & 2\end{array}\right]\left[\begin{array}{c}160 \\ -100 \\ 360\end{array}\right] \][/tex]
Thus, the correct answer is the first option.
[tex]\[ \begin{array}{l} x + y + z = 160 \\ x - 2y - z = -100 \\ 2x + 3y + 2z = 360 \\ \end{array} \][/tex]
This system can be expressed in matrix form as [tex]\( A \mathbf{x} = \mathbf{B} \)[/tex], where:
[tex]\[ A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & -2 & -1 \\ 2 & 3 & 2 \end{bmatrix}, \quad \mathbf{x} = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, \quad \text{and} \quad B = \begin{bmatrix} 160 \\ -100 \\ 360 \end{bmatrix} \][/tex]
Next, let's evaluate each option to find the one that represents the solution.
### Option 1:
[tex]\[ \left[\begin{array}{ccc}1 & 1 & 1 \\ 1 & -2 & -1 \\ 2 & 3 & 2\end{array}\right] \left[\begin{array}{c}160 \\ -100 \\ 360\end{array}\right] \][/tex]
Multiplying the matrix [tex]\( A \)[/tex] by vector [tex]\( B \)[/tex] gives [tex]\(A \mathbf{B}\)[/tex]. This operation uniquely resolves to check if [tex]\( A \mathbf{B} \)[/tex] results in [tex]\( B \)[/tex] which makes the system true. Based on the analysis:
[tex]\[ A \begin{bmatrix} 160 \\ -100 \\ 360 \end{bmatrix} = \begin{bmatrix} 160 \\ -100 \\ 360 \end{bmatrix} \][/tex]
Therefore, option 1 provides the correct matrix equation representing the solution.
### Option 2:
[tex]\[ \left[\begin{array}{ccc}0.5 & -0.5 & 0.5 \\ 2 & 0 & -1 \\ 3.5 & 0.5 & 1.5\end{array}\right] \left[\begin{array}{c}360 \\ -100 \\ 160\end{array}\right] \][/tex]
Evaluating this multiplication, it does not result in [tex]\( \mathbf{B} = \begin{bmatrix} 160 \\ -100 \\ 360 \end{bmatrix} \)[/tex].
### Option 3:
[tex]\[ \left[\begin{array}{ccc}-0.5 & 0.5 & 0.5 \\ -2 & 0 & 1 \\ 3.5 & -0.5 & -1.5\end{array}\right] \left[\begin{array}{c}160 \\ -100 \\ 360\end{array}\right] \][/tex]
Similarly, evaluating this multiplication does not result in [tex]\( \mathbf{B} \)[/tex].
### Option 4:
[tex]\[ \left[\begin{array}{ccc}-1 & -1 & -1 \\ -1 & 2 & 1 \\ -2 & -3 & -2\end{array}\right] \left[\begin{array}{c}360 \\ -100 \\ 160\end{array}\right] \][/tex]
Likewise, evaluating this multiplication, does not result in [tex]\( \mathbf{B} \)[/tex].
### Conclusion:
The option that satisfies [tex]\( A \mathbf{x} = \mathbf{B} \)[/tex] and correctly represents the solution to the system of equations is:
[tex]\[ \left[\begin{array}{ccc}1 & 1 & 1 \\ 1 & -2 & -1 \\ 2 & 3 & 2\end{array}\right]\left[\begin{array}{c}160 \\ -100 \\ 360\end{array}\right] \][/tex]
Thus, the correct answer is the first option.
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.
