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.
