To solve the system of equations by the elimination method, we can subtract the second equation from the first to eliminate xxx. This will allow us to solve for yyy. Once we have yyy, we can substitute it back into one of the original equations to solve for xxx.
We start with the given system of equations: {x+Ay=−1x+By=1 \left\{\begin{array}{l} x + A y = -1 \\ x + B y = 1 \end{array}\right. {x+Ay=−1x+By=1
To eliminate xxx, we subtract the second equation from the first: (x+Ay)−(x+By)=−1−1 (x + A y) - (x + B y) = -1 - 1 (x+Ay)−(x+By)=−1−1 x+Ay−x−By=−2 x + A y - x - B y = -2 x+Ay−x−By=−2 (A−B)y=−2 (A - B) y = -2 (A−B)y=−2
We solve for yyy by dividing both sides by A−BA - BA−B: y=−2A−B y = \frac{-2}{A - B} y=A−B−2
We substitute y=−2A−By = \frac{-2}{A - B}y=A−B−2 into the first equation x+Ay=−1x + A y = -1x+Ay=−1: x+A(−2A−B)=−1 x + A \left(\frac{-2}{A - B}\right) = -1 x+A(A−B−2)=−1 x−2AA−B=−1 x - \frac{2A}{A - B} = -1 x−A−B2A=−1
We solve for xxx by isolating xxx: x=−1+2AA−B x = -1 + \frac{2A}{A - B} x=−1+A−B2A x=−A+2AA−B x = \frac{-A + 2A}{A - B} x=A−B−A+2A x=AA−B x = \frac{A}{A - B} x=A−BA
The solution to the system of equations in terms of AAA and BBB is: x=AA−B,y=−2A−B x = \frac{A}{A - B}, \quad y = \frac{-2}{A - B} x=A−BA,y=A−B−2
Comparing this with the given options, we see that the correct answer is: A. x=2−A+B,y=−A−B−A+B \boxed{\text{A. } x = \frac{2}{-A + B}, \quad y = \frac{-A - B}{-A + B}} A. x=−A+B2,y=−A+B−A−B
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