(i) If $\left(\frac{a}{3}+1, b-\frac{2}{3}\right)=\left(\frac{5}{3}, \frac{1}{3}\right)$, find the values of $a$ and $b$.
(ii) If $(x+1,1)=(3, y-2)$, find the values of $x$ and $y$.
(i) $\left(\frac{a}{3}+1, b-\frac{2}{3}\right)=\left(\frac{5}{3}, \frac{1}{3}\right)$
By the definition of equality of ordered pairs, we have:
$\left(\frac{a}{3}+1, b-\frac{2}{3}\right)=\left(\frac{5}{3}, \frac{1}{3}\right)$
$\Rightarrow\left(\frac{a}{3}+1\right)=\frac{5}{3}$ and $\left(b-\frac{2}{3}\right)=\frac{1}{3}$
$\Rightarrow \frac{a}{3}=\frac{5}{3}-1$ and $b=\frac{1}{3}+\frac{2}{3}$
$\Rightarrow \frac{a}{3}=\frac{2}{3}$ and $b=1$
$\Rightarrow a=2$ and $b=1$
(ii) (x + 1, 1) = (3, y − 2)
By the definition of equality of ordered pairs, we have:
$(x+1)=3$ and $1=(y-2)$
$\Rightarrow x=2$ and $y=3$