# Consider a solution x_1 of the linear system Ax=b. Justify the facts stated in parts (a) and (b): a) If x_h is a solution of the system Ax=0, then x_1+x_h is a solution of the system Ax=b. b) If x_2 is another solution of the system Ax=b, then x_2-x_1 is a solution of the system Ax=0

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Consider a solution $$\displaystyle{x}_{{1}}$$ of the linear system Ax=b. Justify the facts stated in parts (a) and (b):
a) If $$\displaystyle{x}_{{h}}$$ is a solution of the system Ax=0, then $$\displaystyle{x}_{{1}}+{x}_{{h}}$$ is a solution of the system Ax=b.
b) If $$\displaystyle{x}_{{2}}$$ is another solution of the system Ax=b, then $$\displaystyle{x}_{{2}}-{x}_{{1}}$$ is a solution of the system Ax=0

2021-03-22
Question:
Consider a solution $$\displaystyle{x}_{{1}}$$ of the linear system Ax=b. Justify the facts stated in parts (a) and (b):
Hence
$$\displaystyle{A}{x}_{{h}}={0}$$
$$\displaystyle{A}{x}_{{1}}+{a}{x}_{{h}}={b}+{0}={b}$$
$$\displaystyle{A}{\left({x}_{{1}}+{x}_{{h}}\right)}={b}$$
Hence
$$\displaystyle{\left({x}_{{1}}+{x}_{{h}}\right)}$$ is asolution of $$\displaystyle{A}{x}={b}$$ since $$\displaystyle{x}={x}_{{1}}+{x}_{{h}}$$ satisfies this equation.
b) If $$\displaystyle{x}_{{2}}$$ is another solution of the system Ax=b, then $$\displaystyle{x}_{{2}}-{x}_{{1}}$$ is asolution of the system Ax=0
$$\displaystyle{A}{x}_{{2}}={b}$$
$$\displaystyle{A}{x}_{{2}}={A}{x}_{{1}}={b}-{b}={0}$$
$$\displaystyle{A}{\left({x}_{{2}}-{x}_{{1}}\right)}={0}$$
Hence $$\displaystyle{\left({x}_{{2}}-{x}_{{1}}\right)}$$ is solution of
$$\displaystyle{A}{x}={0}$$
Since $$\displaystyle{x}={x}_{{2}}-{x}_{{1}}$$ satisfies the equation.
Totally puzzled here