Prove algebraically that $\quad 3.4 \dot{7}=\frac{313}{90}$

Step 1 ：Let's represent the repeating decimal $3.4\dot{7}$ as a variable, say $x$. So, $x = 3.4\dot{7}$

Step 2 ：To get rid of the repeating decimal, we can multiply both sides of the equation by 10: $10x = 34.\dot{7}$

Step 3 ：Now, subtract the original equation from the new equation: $10x - x = 34.\dot{7} - 3.4\dot{7}$

Step 4 ：This simplifies to: $9x = 31$

Step 5 ：Now, divide both sides by 9 to solve for $x$: $x = \frac{31}{9}$

Step 6 ：However, we need to express the result as a mixed number: $x = 3 + \frac{4}{9}$

Step 7 ：Now, we can convert the mixed number to an improper fraction: $x = \frac{3 \cdot 9 + 4}{9} = \frac{27 + 4}{9}$

Step 8 ：Simplify the fraction: $x = \frac{31}{9}$

Step 9 ：So, $3.4\dot{7} = \boxed{\frac{31}{9}}$