To form the augmented matrix that represents the system of equations given:
1. [tex]\[ y = 10 \][/tex]
2. [tex]\[ 4x - 5y = 3 \][/tex]
We need to express each equation in a way that aligns the coefficients of [tex]\( x \)[/tex], [tex]\( y \)[/tex], and the constants on the right-hand side.
1. Rewriting the first equation [tex]\( y = 10 \)[/tex]:
[tex]\[ 0x + 1y = 10 \][/tex]
This indicates that the coefficient of [tex]\( x \)[/tex] is 0, the coefficient of [tex]\( y \)[/tex] is 1, and the constant term is 10.
2. The second equation [tex]\( 4x - 5y = 3 \)[/tex] already has the coefficients and constants clearly defined as:
[tex]\[ 4x + (-5)y = 3 \][/tex]
Here, the coefficient of [tex]\( x \)[/tex] is 4, the coefficient of [tex]\( y \)[/tex] is -5, and the constant term is 3.
Next, we assemble these coefficients into the augmented matrix form:
[tex]\[ \begin{array}{ccc|c}
& x & y & \\
\hline
0 & 1 & | & 10 \\
4 & -5 & | & 3 \\
\end{array} \][/tex]
This translates to:
[tex]\[ \left[\begin{array}{ccc}
0 & 1 & 10 \\
4 & -5 & 3
\end{array}\right] \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{\left[\begin{array}{ccc}0 & 1 & 10 \\ 4 & -5 & 3\end{array}\right]} \][/tex]
Hence, the correct choice is:
C. [tex]\(\left[\begin{array}{ccc}0 & 1 & 10 \\ 4 & -5 & 3\end{array}\right]\)[/tex].
