To write the given system of equations as an augmented matrix, we need to express the system in matrix form, where each row of the matrix represents one equation. The elements of the matrix will correspond to the coefficients of the variables in the equations and the constants on the right-hand side of the equations.
Given the system:
[tex]\[
\left\{
\begin{aligned}
n + 9m - s &= 200, \\
m &= 300, \\
-4m + s &= 400
\end{aligned}
\right.
\][/tex]
We will create an augmented matrix where each row includes the coefficients of the variables [tex]\( n \)[/tex], [tex]\( m \)[/tex], and [tex]\( s \)[/tex], followed by the constant term on the right-hand side of the equation.
1. For the first equation [tex]\( n + 9m - s = 200 \)[/tex], the coefficients are [tex]\( 1, 9, -1 \)[/tex] and the constant is [tex]\( 200 \)[/tex].
2. For the second equation [tex]\( m = 300 \)[/tex], the coefficients are [tex]\( 0, 1, 0 \)[/tex] and the constant is [tex]\( 300 \)[/tex] since there is no [tex]\( n \)[/tex] term (coefficient is [tex]\( 0 \)[/tex]) and no [tex]\( s \)[/tex] term (coefficient is [tex]\( 0 \)[/tex]).
3. For the third equation [tex]\( -4m + s = 400 \)[/tex], the coefficients are [tex]\( 0, -4, 1 \)[/tex] and the constant is [tex]\( 400 \)[/tex] since there is no [tex]\( n \)[/tex] term (coefficient is [tex]\( 0 \)[/tex]).
Let us now write the augmented matrix:
[tex]\[
\begin{pmatrix}
1 & 9 & -1 & | & 200 \\
0 & 1 & 0 & | & 300 \\
0 & -4 & 1 & | & 400
\end{pmatrix}
\][/tex]
For simplicity, we can drop the vertical line that separates the coefficient matrix from the constants:
[tex]\[
\begin{pmatrix}
1 & 9 & -1 & 200 \\
0 & 1 & 0 & 300 \\
0 & -4 & 1 & 400
\end{pmatrix}
\][/tex]
This is the augmented matrix that represents the given system of equations:
[tex]\[
\begin{pmatrix}
1 & 9 & -1 & 200 \\
0 & 1 & 0 & 300 \\
0 & -4 & 1 & 400
\end{pmatrix}
\][/tex]
