To determine which system of equations led to the intermediate equation [tex]\(9x = 27\)[/tex], let’s analyze each of the four given systems step by step:
### First system:
[tex]\[
\begin{array}{l}
10x - y = 15 \\
x + y = -12
\end{array}
\][/tex]
1. Step 1: Add the equations to eliminate [tex]\( y \)[/tex].
[tex]\[
\begin{array}{r}
10x - y + x + y = 15 + (-12) \\
11x = 3 \\
\end{array}
\][/tex]
Since this step doesn’t match the intermediate step [tex]\(9x = 27\)[/tex], we eliminate this system.
### Second system:
[tex]\[
\begin{array}{l}
-9x - 2y = 21 \\
7x - 2y = 15
\end{array}
\][/tex]
1. Step 1: Subtract the second equation from the first to eliminate [tex]\( y \)[/tex].
[tex]\[
\begin{array}{r}
-9x - 2y - (7x - 2y) = 21 - 15 \\
-16x = 6 \\
x = -\frac{6}{16} \\
\end{array}
\][/tex]
This does not simplify to [tex]\(9x = 27\)[/tex], so this system is also incorrect.
### Third system:
[tex]\[
\begin{array}{l}
x + y = 6 \\
4x + 3y = 24
\end{array}
\][/tex]
1. Step 1: Solve the first equation for [tex]\( y \)[/tex].
[tex]\[
\begin{array}{r}
y = 6 - x
\end{array}
\][/tex]
2. Step 2: Substitute [tex]\( y = 6 - x \)[/tex] into the second equation.
[tex]\[
\begin{array}{r}
4x + 3(6 - x) = 24 \\
4x + 18 - 3x = 24 \\
x + 18 = 24 \\
x = 6
\end{array}
\][/tex]
3. Step 3: Substitute [tex]\( x = 6 \)[/tex] back into the first equation.
[tex]\[
\begin{array}{r}
6 + y = 6 \\
y = 0 \\
\end{array}
\][/tex]
Checking if this gives [tex]\(9x = 27\)[/tex].
[tex]\[
\begin{array}{r}
9 \cdot 6 = 54 \\
\end{array}
\][/tex]
So, this does not lead to the intermediate step [tex]\(9x = 27\)[/tex].
### Fourth system:
[tex]\[
\begin{array}{l}
-5x - 3y = 3 \\
9x = 27
\end{array}
\][/tex]
1. Step 1: The second equation directly is:
[tex]\[
9x = 27
\][/tex]
2. Step 2: Solve for [tex]\( x \)[/tex].
[tex]\[
x = \frac{27}{9} = 3
\][/tex]
3. Step 3: Substitute [tex]\( x = 3 \)[/tex] back to find [tex]\( y \)[/tex].
[tex]\[
\begin{array}{r}
-5(3) - 3y = 3 \\
-15 - 3y = 3 \\
-3y = 18 \\
y = -6
\end{array}
\][/tex]
Thus, this matches the intermediate step [tex]\(9x = 27\)[/tex].
Therefore, the answer is the fourth system:
[tex]\[
\begin{array}{l}
-5x - 3y = 3 \\
9x = 27
\end{array}
\][/tex]
Based on the analysis, the intermediate step [tex]\(9x = 27\)[/tex] is derived from the fourth system of equations.
