what does the relation $Ax+b=A'x+b', \ x \in \mathbb{R}^2$ mean?

Question

Let $A,A' \in GL_2(\mathbb{R})$ and $b,b' \in \mathbb{R}^2$, then what does the relation $Ax+b=A'x+b', \ x \in \mathbb{R}^2$ mean ?

Can I conclude $A=A'$ and $b=b'$ ?

Answer 1

If $Ax+b=A'x+b'$ holds for all $x \in \mathbb R^2$, then, with $x=0$, we get $b=b'.$ Hence $Ax=A'x$ holds for all $x.$ It follows that

$$A=A'.$$