Answer :
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.
### 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.
