To determine which of the equations has infinitely many solutions, we need to simplify each equation and check if they are equivalent or lead to a tautology.
Let's start with each equation one by one.
Equation 1:
\[ 6(x + 3) + x = 7x + 2 + 1 \] Simplifying the left side: \[ 6x + 18 + x = 7x + 3 \implies 7x + 18 = 7x + 3 \] Subtracting \(7x\) from both sides: \[ 18 = 3 \] This is a contradiction.
Equation 2:
\[ 6(x + 3) + x = 7x + 5 \] Simplifying the left side: \[ 6x + 18 + x = 7x + 5 \implies 7x + 18 = 7x + 5 \] Subtracting \(7x\) from both sides: \[ 18 = 5 \] This is a contradiction.
Equation 3:
\[ 6(x + 3) + x = 6x + 3 + 15 \] Simplifying the left side: \[ 6x + 18 + x = 6x + 3 + 15 \implies 7x + 18 = 6x + 18 \] Subtracting \(6x + 18\) from both sides: \[ x = 0 \] This has a single solution.
Equation 4:
\[ 6(x + 3) + x = 7x + 9 + 9 \] Simplifying the left side: \[ 6x + 18 + x = 7x + 18 \implies 7x + 18 = 7x + 18 \] This simplifies to a tautology since both sides are equal for all values of \(x\).
Conclusion:
The fourth equation, \(6(x + 3) + x = 7x + 9 + 9\), has infinitely many solutions because it is true for all \(x\).