Question

System a
-x + y = 1
X + y = 11

System b

X + y = 11
?

What is the true equivalent system
2x + 2y = 12

2y = 12

-2x - 2y = -22

-x - y = -1

Answer 1

To determine which of the options given represents an equivalent system to the systems you provided, we first need to analyze the systems.

### System A:
1. \(-x + y = 1\) (Let's represent it as Equation 1)
2. \(x + y = 11\) (Let's represent it as Equation 2)

### From System A:
We can derive several equations from these two equations to check for equivalence.

1. Rearranging Equation 1:
\[ y = x + 1 \]

2. Substituting \(y\) from Equation 1 into Equation 2:
\[ x + (x + 1) = 11 \]
\[ 2x + 1 = 11 \]
\[ 2x = 10 \]
\[ x = 5 \]

Using \(x = 5\) in Equation 2:
\[ 5 + y = 11 \]
\[ y = 6 \]

So the solution to System A is \(x = 5\) and \(y = 6\).

### System B:
1. \(x + y = 11\)

We only have one equation, which is also the second equation of System A.

### Now let’s analyze the equivalent systems:
1. **\(2x + 2y = 12\)**:
- This can be simplified to \(x + y = 6\), which is **not equivalent**.

2. **\(2y = 12\)**:
- This simplifies to \(y = 6\). Substituting this into \(x + y = 11\) gives \(x + 6 = 11\), which results in \(x = 5\). So this is equivalent.

3. **\(-2x - 2y = -22\)**:
- This can be simplified to \(x + y = 11\), which is equivalent to one of the original equations.

4. **\(-x - y = -1\)**:
- This can be rearranged to \(x + y = 1\), which is **not equivalent**.

### Conclusion:
The true equivalent systems from the options provided are:

- **\(2y = 12\)**
- **\(-2x - 2y = -22\)**
- **\(x + y = 11\)** (Already present in System B)

So, the answer is that \(2y = 12\) and \(-2x - 2y = -22\) are valid equivalent systems to the original system.