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To write a system of equations from the given augmented matrix, we need to understand what an augmented matrix represents. An augmented matrix includes the coefficients of the variables from a system of linear equations and the constants on the right side of the equations.
Given the augmented matrix:
[tex]\[ \left[\begin{array}{cc|c} 4 & -2 & 3 \\ 6 & 11 & 9 \end{array}\right] \][/tex]
We can decipher this into a system of equations:
### Step 1: Identify the components
The augmented matrix consists of:
- The coefficients of the first equation in the first row: [tex]\([4, -2]\)[/tex]
- The constant term of the first equation: [tex]\(3\)[/tex]
- The coefficients of the second equation in the second row: [tex]\([6, 11]\)[/tex]
- The constant term of the second equation: [tex]\(9\)[/tex]
### Step 2: Write out the equations
Using these components, we can write out the system of linear equations:
1. The first row corresponds to:
[tex]\[ 4x - 2y = 3 \][/tex]
2. The second row corresponds to:
[tex]\[ 6x + 11y = 9 \][/tex]
### Step 3: Compile the system
The overall system of linear equations represented by the given augmented matrix is:
[tex]\[ \begin{cases} 4x - 2y = 3 \\ 6x + 11y = 9 \end{cases} \][/tex]
This system of equations can be solved using various methods such as substitution, elimination, or matrix operations, but the equations are correctly derived directly from the augmented matrix.
Given the augmented matrix:
[tex]\[ \left[\begin{array}{cc|c} 4 & -2 & 3 \\ 6 & 11 & 9 \end{array}\right] \][/tex]
We can decipher this into a system of equations:
### Step 1: Identify the components
The augmented matrix consists of:
- The coefficients of the first equation in the first row: [tex]\([4, -2]\)[/tex]
- The constant term of the first equation: [tex]\(3\)[/tex]
- The coefficients of the second equation in the second row: [tex]\([6, 11]\)[/tex]
- The constant term of the second equation: [tex]\(9\)[/tex]
### Step 2: Write out the equations
Using these components, we can write out the system of linear equations:
1. The first row corresponds to:
[tex]\[ 4x - 2y = 3 \][/tex]
2. The second row corresponds to:
[tex]\[ 6x + 11y = 9 \][/tex]
### Step 3: Compile the system
The overall system of linear equations represented by the given augmented matrix is:
[tex]\[ \begin{cases} 4x - 2y = 3 \\ 6x + 11y = 9 \end{cases} \][/tex]
This system of equations can be solved using various methods such as substitution, elimination, or matrix operations, but the equations are correctly derived directly from the augmented matrix.
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