1. The problem asks to find the value of $x$ in the equation $$\frac{3x+1}{2} = 7.$$\n\n2. The formula used here is to solve linear equations by isolating the variable $x$.\n\n3. First, multiply both sides of the equation by 2 to eliminate the denominator:\n$$2 \times \frac{3x+1}{2} = 2 \times 7$$\nThis simplifies to:\n$$\cancel{2} \times \frac{3x+1}{\cancel{2}} = 14$$\n$$3x + 1 = 14.$$\n\n4. Next, subtract 1 from both sides to isolate the term with $x$:\n$$3x + 1 - 1 = 14 - 1$$\n$$3x = 13.$$\n\n5. Finally, divide both sides by 3 to solve for $x$:\n$$\frac{3x}{3} = \frac{13}{3}$$\n$$\cancel{3}x / \cancel{3} = \frac{13}{3}$$\n$$x = \frac{13}{3}.$$\n\n6. Therefore, the solution is $x = \frac{13}{3}$ or approximately $4.33$.