To determine which equation creates a true equivalent system, we need to analyze the equations in the systems.
First, let's rewrite System A:
- \(-x + y = 1\)
- \(x + y = 11\)
From the second equation in System A (\(x + y = 11\)), we can express \(y\) as follows:
\[ y = 11 - x \]
Next, we can substitute \(y\) from the second equation into the first equation to check if both equations are consistent:
\[ -x + (11 - x) = 1 \]
This simplifies to:
\[ -x + 11 - x = 1 \] \[ -2x + 11 = 1 \] \[ -2x = 1 - 11 \] \[ -2x = -10 \] \[ x = 5 \]
Now substituting \(x = 5\) back into the second equation \(x + y = 11\):
\[ 5 + y = 11 \] \[ y = 6 \]
So the solution of System A is \( (x, y) = (5, 6) \).
Now consider System B, which has only one equation:
\[ x + y = 11 \]
This means any equivalent equation should yield consistent results with the original system.
Now, we can check each response option to see which one matches with either the first equation while maintaining the relationship between \(x\) and \(y\).
- \(2x + 2y = 12\)
Dividing through by 2 gives:
\[ x + y = 6 \]
This is not equivalent to the given equations as it does not match.
- \(-2x - 2y = -22\)
Dividing through by -2 gives:
\[ x + y = 11 \]
This is indeed equivalent to the equation in System B.
- \(-x - y = -1\)
By rearranging, we find it gives:
\[ x + y = 1 \]
This does not match either.
- \(2y = 12\)
Dividing gives:
\[ y = 6 \]
However, we still need to express \(x\) as well, so this is not establishing an equivalent system overall.
Thus, the correct response that creates a true equivalent system is:
\(\boxed{-2x - 2y = -22}\)
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