Answer :
To represent the given system of equations in matrix form, we start by identifying the coefficients of the variables and the constants from each equation. The system of equations is:
[tex]\[ \left\{\begin{array}{l} x + 2y = 11 \\ 2x + 3y = 18 \end{array}\right. \][/tex]
### Step 1: Matrix of Coefficients
We need to form a coefficient matrix [tex]\(A\)[/tex] from the coefficients of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] in both equations.
For the first equation [tex]\(x + 2y = 11\)[/tex], the coefficients are [tex]\(1\)[/tex] (for [tex]\(x\)[/tex]) and [tex]\(2\)[/tex] (for [tex]\(y\)[/tex]).
For the second equation [tex]\(2x + 3y = 18\)[/tex], the coefficients are [tex]\(2\)[/tex] (for [tex]\(x\)[/tex]) and [tex]\(3\)[/tex] (for [tex]\(y\)[/tex]).
Thus, the coefficient matrix [tex]\(A\)[/tex] is:
[tex]\[ A = \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix} \][/tex]
### Step 2: Variable Matrix
The variables involved are [tex]\(x\)[/tex] and [tex]\(y\)[/tex]. We represent them as a column matrix [tex]\(X\)[/tex]:
[tex]\[ X = \begin{bmatrix} x \\ y \end{bmatrix} \][/tex]
### Step 3: Constant Matrix
The constants on the right-hand side of the equations are 11 and 18. We represent them as a column matrix [tex]\(B\)[/tex]:
[tex]\[ B = \begin{bmatrix} 11 \\ 18 \end{bmatrix} \][/tex]
### Step 4: Forming the Matrix Equation
We combine the above elements to form the matrix equation [tex]\(AX = B\)[/tex], which represents the system of equations:
[tex]\[ \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 11 \\ 18 \end{bmatrix} \][/tex]
### Conclusion
The matrix equation that correctly represents the given system of equations is:
[tex]\[ \left[\begin{array}{ll}1 & 2 \\ 2 & 3\end{array}\right]\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{l}11 \\ 18\end{array}\right] \][/tex]
Thus, the correct choice from the given options is:
[tex]\[ \left[\begin{array}{ll}1 & 2 \\ 2 & 3\end{array}\right]\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{l}11 \\ 18\end{array}\right] \][/tex]
[tex]\[ \left\{\begin{array}{l} x + 2y = 11 \\ 2x + 3y = 18 \end{array}\right. \][/tex]
### Step 1: Matrix of Coefficients
We need to form a coefficient matrix [tex]\(A\)[/tex] from the coefficients of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] in both equations.
For the first equation [tex]\(x + 2y = 11\)[/tex], the coefficients are [tex]\(1\)[/tex] (for [tex]\(x\)[/tex]) and [tex]\(2\)[/tex] (for [tex]\(y\)[/tex]).
For the second equation [tex]\(2x + 3y = 18\)[/tex], the coefficients are [tex]\(2\)[/tex] (for [tex]\(x\)[/tex]) and [tex]\(3\)[/tex] (for [tex]\(y\)[/tex]).
Thus, the coefficient matrix [tex]\(A\)[/tex] is:
[tex]\[ A = \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix} \][/tex]
### Step 2: Variable Matrix
The variables involved are [tex]\(x\)[/tex] and [tex]\(y\)[/tex]. We represent them as a column matrix [tex]\(X\)[/tex]:
[tex]\[ X = \begin{bmatrix} x \\ y \end{bmatrix} \][/tex]
### Step 3: Constant Matrix
The constants on the right-hand side of the equations are 11 and 18. We represent them as a column matrix [tex]\(B\)[/tex]:
[tex]\[ B = \begin{bmatrix} 11 \\ 18 \end{bmatrix} \][/tex]
### Step 4: Forming the Matrix Equation
We combine the above elements to form the matrix equation [tex]\(AX = B\)[/tex], which represents the system of equations:
[tex]\[ \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 11 \\ 18 \end{bmatrix} \][/tex]
### Conclusion
The matrix equation that correctly represents the given system of equations is:
[tex]\[ \left[\begin{array}{ll}1 & 2 \\ 2 & 3\end{array}\right]\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{l}11 \\ 18\end{array}\right] \][/tex]
Thus, the correct choice from the given options is:
[tex]\[ \left[\begin{array}{ll}1 & 2 \\ 2 & 3\end{array}\right]\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{l}11 \\ 18\end{array}\right] \][/tex]
