(i) If $left( rac{a}{3}+1, b- rac{2}{3} ight)=left( rac{5}{3}, rac{1}{3} ight)$, find the values of $a$ and $b$.

(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 $y=3$