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The matrix below is the final matrix form for a system of two linear equations in the variables x₁ and x2. Write the solution of the system. 1 0-8 01 6 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. and x₂ = O A. The unique solution to the system is x₁ = OB. There are infinitely many solutions. The solution is x₁ = OC. There is no solution. and x₂ =t, for any real number t. (Type an expression using t as the variable.)

The matrix below is the final matrix form for a system of two linear equations in the variables x₁ and x2. Write the solution of the system. 1 0-8 01 6 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. and x₂ = O A. The unique solution to the system is x₁ = OB. There are infinitely many solutions. The solution is x₁ = OC. There is no solution. and x₂ =t, for any real number t. (Type an expression using t as the variable.)

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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### Solving a System of Linear Equations Using Matrix Representation The matrix below is the final matrix form for a system of two linear equations in the variables \( x_1 \) and \( x_2 \). Write the solution of the system. \[ \begin{bmatrix} 1 & 0 & | & -8\\ 0 & 1 & | & 6 \end{bmatrix} \] #### System of Equations Interpretation This matrix represents the following system of equations: 1. \( x_1 = -8 \) 2. \( x_2 = 6 \) #### Solution Choices Select the correct choice below and, if necessary, fill in the answer box to complete your choice. - **A.** The unique solution to the system is \( x_1 = \) [input box] and \( x_2 = \) [input box]. - **B.** There are infinitely many solutions. The solution is \( x_1 = \) [input box] and \( x_2 = t \), for any real number \( t \). (Type an expression using \( t \) as the variable.) - **C.** There is no solution. #### Explanation For the given matrix, the corresponding values for \( x_1 \) and \( x_2 \) are directly derived from the matrix form. The unique solution to this system is: - \( x_1 = -8 \) - \( x_2 = 6 \) Therefore, the correct choice is **A**. The unique solution to the system is \( x_1 = -8 \) and \( x_2 = 6 \).
Transcribed Image Text:### Solving a System of Linear Equations Using Matrix Representation The matrix below is the final matrix form for a system of two linear equations in the variables \( x_1 \) and \( x_2 \). Write the solution of the system. \[ \begin{bmatrix} 1 & 0 & | & -8\\ 0 & 1 & | & 6 \end{bmatrix} \] #### System of Equations Interpretation This matrix represents the following system of equations: 1. \( x_1 = -8 \) 2. \( x_2 = 6 \) #### Solution Choices Select the correct choice below and, if necessary, fill in the answer box to complete your choice. - **A.** The unique solution to the system is \( x_1 = \) [input box] and \( x_2 = \) [input box]. - **B.** There are infinitely many solutions. The solution is \( x_1 = \) [input box] and \( x_2 = t \), for any real number \( t \). (Type an expression using \( t \) as the variable.) - **C.** There is no solution. #### Explanation For the given matrix, the corresponding values for \( x_1 \) and \( x_2 \) are directly derived from the matrix form. The unique solution to this system is: - \( x_1 = -8 \) - \( x_2 = 6 \) Therefore, the correct choice is **A**. The unique solution to the system is \( x_1 = -8 \) and \( x_2 = 6 \).
**Solving a System of Linear Equations Using Matrix Form** The matrix below is the final matrix form for a system of two linear equations in the variables \( x_1 \) and \( x_2 \). Write the solution of the system. \[ \begin{bmatrix} 1 & -2 & \vert & 13 \\ 0 & 0 & \vert & 0 \end{bmatrix} \] --- **Question:** Select the correct choice below and, if necessary, fill in the answer box to complete your choice. **A.** The unique solution to the system is \( x_1 = \) [ ] and \( x_2 = \) [ ]. **B.** There are infinitely many solutions. The solution is \( x_1 = \) [ ] and \( x_2 = t \), for any real number \( t \). (Type an expression using \( t \) as the variable.) **C.** There is no solution. --- **Explanation:** The given matrix represents a system of linear equations. The first row translates to: \[ x_1 - 2x_2 = 13 \] The second row is: \[ 0 = 0 \] The second equation is true for all values of \( x_1 \) and \( x_2 \). Therefore, it does not provide any additional constraints on the values of \( x_1 \) and \( x_2 \). This indicates that there are infinitely many solutions where \( x_2 \) (let's call it \( t \)) can be any real number. Consequently, \( x_1 \) can be expressed in terms of \( t \) from the first equation. That is, for \( x_2 = t \): \[ x_1 - 2t = 13 \] \[ x_1 = 13 + 2t \] Thus, the solution is: \[ x_1 = 13 + 2t \] \[ x_2 = t \] Therefore, the correct choice is: **B.** There are infinitely many solutions. The solution is \( x_1 = 13 + 2t \) and \( x_2 = t \), for any real number \( t \).
Transcribed Image Text:**Solving a System of Linear Equations Using Matrix Form** The matrix below is the final matrix form for a system of two linear equations in the variables \( x_1 \) and \( x_2 \). Write the solution of the system. \[ \begin{bmatrix} 1 & -2 & \vert & 13 \\ 0 & 0 & \vert & 0 \end{bmatrix} \] --- **Question:** Select the correct choice below and, if necessary, fill in the answer box to complete your choice. **A.** The unique solution to the system is \( x_1 = \) [ ] and \( x_2 = \) [ ]. **B.** There are infinitely many solutions. The solution is \( x_1 = \) [ ] and \( x_2 = t \), for any real number \( t \). (Type an expression using \( t \) as the variable.) **C.** There is no solution. --- **Explanation:** The given matrix represents a system of linear equations. The first row translates to: \[ x_1 - 2x_2 = 13 \] The second row is: \[ 0 = 0 \] The second equation is true for all values of \( x_1 \) and \( x_2 \). Therefore, it does not provide any additional constraints on the values of \( x_1 \) and \( x_2 \). This indicates that there are infinitely many solutions where \( x_2 \) (let's call it \( t \)) can be any real number. Consequently, \( x_1 \) can be expressed in terms of \( t \) from the first equation. That is, for \( x_2 = t \): \[ x_1 - 2t = 13 \] \[ x_1 = 13 + 2t \] Thus, the solution is: \[ x_1 = 13 + 2t \] \[ x_2 = t \] Therefore, the correct choice is: **B.** There are infinitely many solutions. The solution is \( x_1 = 13 + 2t \) and \( x_2 = t \), for any real number \( t \).
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