Skill: Identify equations with infinitely many solutions

To identify an equation with infinitely many solutions:

1. Use the distributive property to remove parentheses.
2. Combine like terms on each side.
3. Compare the simplified sides:
   • If you get a true identity such as $0=0$, there are infinitely many solutions.
   • If you get a false statement such as $0=5$, there is no solution.

Example

Determine the number of solutions:

$$4(x-1)+2x=6x-4$$

Distribute $4$ on the left:

$$4x-4+2x=6x-4$$

Combine like terms:

$$6x-4=6x-4$$

Subtract $6x$ from both sides:

$$-4=-4$$

This is always true. Equivalently, simplifying both sides would produce $0=0$. Therefore, every real value of $x$ makes the equation true.

$$\boxed{\text{The equation has infinitely many solutions.}}$$