Math Problem Statement
Write a vector equation that is equivalent to the system of equations 2 x subscript 1 space plus thin space 4 x subscript 2 space plus thin space 2 x subscript 3 space minus space 6 x subscript 4 space equals space 2 x subscript 1 plus space 2 x subscript 2 space plus space 3 x subscript 3 space minus space 9 x subscript 4 space equals space 5 A. 2 x with rightwards harpoon with barb upwards on top space plus thin space 4 y with rightwards harpoon with barb upwards on top space plus thin space 2 z with rightwards harpoon with barb upwards on top space minus space 6 w with rightwards harpoon with barb upwards on top space equals space 2 B. open square brackets table row cell 2 x subscript 1 end cell row cell x subscript 1 end cell end table close square brackets space plus space open square brackets table row cell 4 x subscript 2 end cell row cell 2 x subscript 2 end cell end table close square brackets space plus thin space open square brackets table row cell 2 x subscript 3 end cell row cell 3 x subscript 3 end cell end table close square brackets space plus space open square brackets table row cell negative 6 x subscript 4 end cell row cell negative 9 x subscript 4 end cell end table close square brackets space equals 2 C. open square brackets table row 2 4 2 cell negative 6 end cell 2 row 1 2 3 cell negative 9 end cell 5 end table close square brackets D. open square brackets table row 2 4 2 cell negative 6 end cell row 1 2 3 cell negative 9 end cell end table close square brackets open square brackets table row cell x subscript 1 end cell row cell x subscript 2 end cell row cell x subscript 3 end cell row cell x subscript 4 end cell end table close square brackets space equals open square brackets table row 2 row 5 end table close square brackets E. x subscript 1 open square brackets table row 2 row 1 end table close square brackets space plus space x subscript 2 open square brackets table row 4 row 2 end table close square brackets space plus thin space x subscript 3 open square brackets table row 2 row 3 end table close square brackets space plus space x subscript 4 open square brackets table row cell negative 6 end cell row cell negative 9 end cell end table close square brackets space equals open square brackets table row 2 row 5 end table close square brackets F. 2 open square brackets table row cell x subscript 1 end cell row cell x subscript 2 end cell end table close square brackets space plus space 4 open square brackets table row cell x subscript 1 end cell row cell x subscript 2 end cell end table close square brackets space plus thin space 2 open square brackets table row cell x subscript 1 end cell row cell x subscript 2 end cell end table close square brackets space minus space 6 open square brackets table row cell x subscript 1 end cell row cell x subscript 2 end cell end table close square brackets space equals space open square brackets table row 2 row 5 end table close square brackets
Solution
To express the given system of equations as a vector equation, we will first organize the terms systematically and then represent them in matrix-vector form. The system is as follows:
The goal is to write these as a vector equation of the form:
Where:
- is the matrix of coefficients,
- is the vector of unknowns ,
- is the vector of constants.
Step-by-step formulation:
We can write the system of equations as:
2 & 4 & 2 & -6 \\ 2 & 2 & 3 & -9 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix} = \begin{bmatrix} 2 \\ 5 \end{bmatrix}$$ Thus, the vector equation corresponding to the system is: $$\mathbf{A} \mathbf{x} = \mathbf{b}$$ Where: $$\mathbf{A} = \begin{bmatrix} 2 & 4 & 2 & -6 \\ 2 & 2 & 3 & -9 \end{bmatrix}, \quad \mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} 2 \\ 5 \end{bmatrix}$$ This is the vector equation that represents the system. ### Answer: The correct vector equation is: $$\begin{bmatrix} 2 & 4 & 2 & -6 \\ 2 & 2 & 3 & -9 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix} = \begin{bmatrix} 2 \\ 5 \end{bmatrix}$$ --- Would you like further explanation or any specific detail on any step of the process? Here are some related questions for you: 1. How do you derive a vector equation from a system of equations? 2. How do we solve vector equations using matrix inversion or Gaussian elimination? 3. Can you explain the meaning of the matrix $$\mathbf{A}$$ in the context of linear transformations? 4. How do you convert a vector equation to a system of equations? 5. What is the geometric interpretation of the vector equation $$\mathbf{A} \mathbf{x} = \mathbf{b}$$? **Tip:** Always check if the matrix $$\mathbf{A}$$ is invertible before trying to solve the equation using matrix inverse; otherwise, alternative methods like Gaussian elimination or row reduction might be required.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Vector Equations
Matrices
Systems of Linear Equations
Formulas
Vector equation form: A * x = b
Matrix multiplication for systems of equations
Theorems
Matrix-vector multiplication
Representation of systems of linear equations in matrix form
Suitable Grade Level
Grades 10-12