Let Ax=b be a nonhomogenous system of linear equations with the unknown x ∈ |R^n. Assume that X1 ∈ |R^n and X2 ∈ |R^n are both solutions of the nonhomogenous system. Which ONE of the following statements must be true?

a) x1 + x2 is a solution of Ax = 0
b) x1 - x2 is a solution of Ax = 0
c) x1 + x2 is a solution of Ax = b
d) x1 - x2 is a solution of Ax = b

I'm not really sure how to approach this question. My current approach is to make up a system of equations such that Ax = b is nonhomogenous.

So I have:

5x - 2y = 1
8x - 3y = 2

The solutions of (x,y) are (1,2) for both equations.

5(1) - 2(2) = 1
8(1) - 3(2) = 1

Not sure what to do next, if I plug in ( x , y ) for ( x1 , x2 ) in the answer choices, none of them seem to be consistent.

  • $\begingroup$ Note that $A(x_1-x_2)=Ax_1-Ax_2=b-b=0$, so... $\endgroup$ – DiegoMath Nov 9 '14 at 2:10

Since $x_1$ and $x_2$ are solutions to the inhomogenous eqaution $Ax = b$, we have $$ Ax_1 = b $$ $$ Ax_2 = b $$

Since $A$ is linear, we have $$A(x_1+x_2) = Ax_1+Ax_2 = b+b=2b$$ $$A(x_1-x_2) = Ax_1-Ax_2 = b-b=0$$

Only option b is necessarily correct.


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