Elimination was used to solve a system of equations. One of the intermediate steps led to the equation [tex]9x = 27[/tex]. Which of the following systems could have led to this equation?

A.
[tex]\[
\begin{array}{l}
10x - y = 15 \\
x + y = -12 \\
9x + 2y = 21
\end{array}
\][/tex]

B.
[tex]\[
\begin{array}{l}
-9x - 2y = 21 \\
7x - 2y = 15 \\
x + y = 6 \\
4x + 3y = 24 \\
-5x - 3y = 3
\end{array}
\][/tex]



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.