Question

Use the table to answer the question.

System A System B
−x+y=1

x+y=11

x+y=11

?

Which of the following shows the equation that creates a true equivalent system?

(1 point)
Responses

2x+2y=12
2 x plus 2 y equals 12

2y=12
2 y equals 12

−2x−2y=−22
negative 2 x minus 2 y equals negative 22

−x−y=−1

Answer 1

To find an equation that creates a true equivalent system to the given systems, we need to analyze the provided equations.

The two equations given from System A and System B are:

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

Now, eliminate \(y\) from these equations to find an equivalent one.

First, we can manipulate these equations:

1. Rearranging the first equation:
\(y = x + 1\)

2. Substitute \(y\) in the second equation:
\[
x + (x + 1) = 11
\]
\[
2x + 1 = 11
\]
\[
2x = 10
\]
\[
x = 5
\]
Now substitute \(x\) back to find \(y\):
\[
y = 5 + 1 = 6
\]

Thus, one solution is \(x = 5\) and \(y = 6\).

Next, let's check which of the given responses corresponds to the original equations and gives the same solutions.

To determine if the proposed equations are equivalent:

1. **\(2x + 2y = 12\)** - Divide by 2:
\[
x + y = 6 \quad \text{(not equivalent)}
\]

2. **\(2y = 12\)** - Divide by 2:
\[
y = 6 \quad \text{(not equivalent to the original)}
\]

3. **\(-2x - 2y = -22\)** - Divide by -2:
\[
x + y = 11 \quad \text{(this is equivalent to System B)}
\]

4. **\(-x - y = -1\)** - Multiply by -1:
\[
x + y = 1 \quad \text{(not equivalent)}
\]

Thus, the equation that creates a true equivalent system from the given options is:

**\(-2x - 2y = -22\)**.