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

1 answer

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 \(x\).

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