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 \(b\)
Question
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 \(b\).
Solution
Since two ordered pairs are equal, their corresponding components are equal.
\[ \frac{a}{3}+1=\frac{5}{3} \]
\[ \frac{a}{3}=\frac{5}{3}-1 \]
\[ \frac{a}{3}=\frac{5}{3}-\frac{3}{3} \]
\[ \frac{a}{3}=\frac{2}{3} \]
\[ a=2 \]
Now,
\[ b-\frac{2}{3}=\frac{1}{3} \]
\[ b=\frac{1}{3}+\frac{2}{3} \]
\[ b=\frac{3}{3} \]
\[ b=1 \]
Therefore, \[ \boxed{a=2,\ b=1} \]