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:
- \(-x + y = 1\)
- \(x + y = 11\)
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:
-
\(-x - y = -1\): This is not equivalent; it changes the sign of both \(y\) and \(x\).
-
\(2x + 2y = 12\): This is equivalent to the second equation \(x + y = 11\) when multiplied by 2 (it should be \(2x + 2y = 22\), which is not provided).
-
\(-2x - 2y = -22\): This is equivalent; it is simply the first equation multiplied by -2.
-
\(2y = 12\): 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.