Use the table to answer the question.

Question

System A System B
−x+y=1
−
𝑥
+
𝑦
=
1

x+y=11
𝑥
+
𝑦
=
11

x+y=11
𝑥
+
𝑦
=
11

?
?

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

(1 point)
Responses

−x−y=−1
−
𝑥
−
𝑦
=
−
1
negative x minus y equals negative 1

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

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

2y=12

Answer 1

To find an equivalent equation for the system that includes the equations given, we can multiply one of the existing equations by a constant.

Let's analyze the existing equations in System A:

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

We'll focus on the first equation \(-x + y = 1\).

To create equivalent equations, we can multiply the entire equation by the same constant. Here are the responses from the options given:

1. \(-x - y = -1\): This is not equivalent; it changes the sign of both \(y\) and \(x\).

2. \(2x + 2y = 12\): This is equivalent to the second equation \(x + y = 11\) when multiplied by 2 (it should be \(2x + 2y = 22\), which is not provided).

3. \(-2x - 2y = -22\): This is equivalent; it is simply the first equation multiplied by -2.

4. \(2y = 12\): This does not directly follow from the given equations.

Among these options, the correct response that creates a true equivalent system to one of the existing equations is:

(-2x - 2y = -22.

This option maintains the validity of the original equations by being a scalar multiple of one of them.