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

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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

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:
\[
6x + 16x_1 + x_2 - 3x + 33x_3 = 77
\]

#### Equation 2:
\[
2x + 22x_2 + 9x + 39x_3 = 0
\]

### 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:
\[
\begin{bmatrix}
6 & 16 & -3 \\
0 & 2 & 9
\end{bmatrix}
\begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix}
=
\begin{bmatrix}
77 \\
0
\end{bmatrix}
\]

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.

Write the system first as a vector equation and then as a matrix equation. 6x 16x1 plus x 2x2 minus 3x 33x3 equals 77. 2x 22x2 plus 9x 39x3 equals 0.

Learn how to represent a system of linear equations using vector and matrix equations. This problem demonstrates converting a system of equations into vector and matrix forms, exploring core concepts like matrix representation and solving systems using Gaussian elimination. Suitable for Grades 10-12 students.

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