Question

Dante wrote several equations and determined that only one of the equations has infinitely many solutions. Which of these equations has infinitely many solutions?
6 (x + 3) + x = 7 x + 2 + 1
6 (x + 3) + x = 7 x + 5
6 (x + 3) + x = 6 x + 3 + 15
6 (x + 3) + x = 7 x + 9 + 9

Answer 1

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 \(x\).