Math Problem Statement
Write the system first as a vector equation and then as a matrix equation.
6 x 16x1
plus+
x 2x2
minus−
3 x 33x3
equals=
77
2 x 22x2
plus+
9 x 39x3
equals=
0
Question content area bottom
Part 1
Write the system as a vector equation where the first equation of the system corresponds to the first row. Select the correct choice below and fill in any answer boxes to complete your choice.
A.enter your response here Start 3 By 1 Table 1st Row 1st Column x 1 2nd Row 1st Column x 2 3rd Row 1st Column x 3 EndTable equals enter your response here
enter your response hereStart 3 By 1 Table 1st Row 1st Column x 1 2nd Row 1st Column x 2 3rd Row 1st Column x 3 EndTable
x1
x2
x3
equals=enter your response here
B.enter your response here Start 1 By 3 Table 1st Row 1st Column x 1 2nd Column x 2 3rd Column x 3 EndTable equals enter your response here
enter your response hereStart 1 By 3 Table 1st Row 1st Column x 1 2nd Column x 2 3rd Column x 3 EndTable
x1
x2
x3
equals=enter your response here
C.x 1 enter your response here plus x 2 enter your response here plus x 3 enter your response here equals enter your response here
x 1x1enter your response hereplus+x 2x2enter your response hereplus+x 3x3enter your response hereequals=enter your response here
Solution
To tackle this system of equations, we can start by writing it first as a vector equation and then as a matrix equation.
Step 1: The given system
You provided two separate systems of equations, which seem to involve different matrix and vector operations:
Equation 1:
Equation 2:
Step 2: Writing as a vector equation
The goal here is to represent each equation as a vector equation. This is generally done by writing the variables and their coefficients in vector and matrix form.
Equation 1 in vector form:
6 & 16 & -3 \\ 0 & 2 & 9 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 77 \\ 0 \end{bmatrix}$$ So, the first equation becomes: $$6x_1 + 16x_2 - 3x_3 = 77$$ Similarly, for the second equation, we have: $$2x_1 + 22x_2 + 9x_3 = 0$$ #### Vector equation summary: This system can now be expressed as: $$\begin{bmatrix} 6 & 16 & -3 \\ 2 & 22 & 9 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 77 \\ 0 \end{bmatrix}$$ ### Step 3: Writing as a matrix equation In matrix form, the system is written as $$A\mathbf{x} = \mathbf{b}$$, where: - $$A$$ is the matrix of coefficients, - $$\mathbf{x}$$ is the column vector of variables $$x_1, x_2, x_3$$, - $$\mathbf{b}$$ is the column vector representing the constants on the right-hand side of the equations. $$\begin{bmatrix} 6 & 16 & -3 \\ 2 & 22 & 9 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 77 \\ 0 \end{bmatrix}$$ This is the matrix form of your system of equations. --- ### Do you want more details or have any questions? ### Related questions: 1. How do we perform matrix multiplication to verify the solution? 2. How can we solve this matrix equation using Gaussian elimination? 3. What is the determinant of the coefficient matrix, and why is it useful? 4. How can we use Cramer's rule to solve this system? 5. What does the system look like if we add more variables? #### Tip: In a system of equations, expressing the system as a matrix equation helps simplify solving it, especially when dealing with larger systems or multiple variables.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Equations
Matrix Representation
Formulas
Ax = b (Matrix equation form)
Vector representation of linear systems
Theorems
Gaussian Elimination
Cramer's Rule
Suitable Grade Level
Grades 10-12