Question

Which of the following shows the equation that creates a true equivalent system?
System A | System B
-x + y = 1 | x + y = 11
x + y = 11 | BLANK
A. 2x + 2y = 12
B. 2y = 12
C. -2x - 2y = -22
D. -x - y = -1

Answer 1

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:
1. \(-x + y = 1\)
2. \(x + y = 11\)

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 \(y\):
1. 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 \(x + y = 11\):

- **Option A: \(2x + 2y = 12\)**
- Dividing everything by 2 gives \(x + y = 6\). This is **not equivalent**.

- **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.

- **Option C: \(-2x - 2y = -22\)**
- Dividing the entire equation by -2 gives \(x + y = 11\), which is equivalent to System B.

- **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.