To determine which equation creates a true equivalent system with System B, we need to see how we can manipulate the equations of System A to arrive at the equation of System B.
System A is:
- \(-x + y = 1\)
- \(x + y = 11\)
We need to focus on the second equation of System B, \(x + y = 11\), and figure out which of the options represents an equivalent equation.
First, let's solve one of the equations in System A for \(y\):
- From \(-x + y = 1\), we can rearrange to get \(y = x + 1\).
Next, we can substitute \(y\) in the second equation of System B: \(x + (x + 1) = 11\) This simplifies to: \(2x + 1 = 11\) So, \(2x = 10\) Thus, \(x = 5\).
Now, we find \(y\): Substituting \(x = 5\) into the equation \(y = x + 1\): \(y = 5 + 1 = 6\). So, the solution for System B is \(x = 5\), \(y = 6\) which we derived from System A.
Now, let's consider the options provided one by one in relation to \(x + y = 11\):
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Option A: \(2x + 2y = 12\)
- Dividing everything by 2 gives \(x + y = 6\). This is not equivalent.
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Option B: \(2y = 12\)
- Dividing by 2 gives \(y = 6\). We need to check \(y\) against \(x + y = 11\) (so \(x + 6 = 11\)), which indicates \(x = 5\). Thus this option could keep the system equivalent implicitly, but it doesn't provide both variables together.
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Option C: \(-2x - 2y = -22\)
- Dividing the entire equation by -2 gives \(x + y = 11\), which is equivalent to System B.
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Option D: \(-x - y = -1\)
- Multiplying by -1 gives \(x + y = 1\), which is not equivalent to \(x + y = 11\).
From this analysis, we find that the correct answer is: C. \(-2x - 2y = -22\). This equation creates a true equivalent system with System B.