A student solved an equation, as shown below. 1/4(8-12x)=x-2(2x+1) 2-3x=x-4x+2 2-3x=-3x+2 2-3x+3x=-3x+3x+2 2=2 based on the students work, complete the statement below. Use the drop-down menus to make the statement below true. The student solved the equation [correctly, incorrectly] because the original equation has [no solution, one solution, infinitely many solutions]

Answer 1

To analyze the student's solution:

1. The original equation is \( \frac{1}{4}(8 - 12x) = x - 2(2x + 1) \).
2. When simplified, we have \( 2 - 3x = x - 4x + 2 \), which simplifies to \( 2 - 3x = -3x + 2 \).
3. The steps continue to show that \( 2 - 3x + 3x = -3x + 3x + 2 \), resulting in \( 2 = 2 \).

Since the final equation \( 2 = 2 \) is always true and is an identity, this means the original equation has infinitely many solutions. The student correctly simplified the equation, leading to this true statement.

Therefore, we can complete the statement as follows:

The student solved the equation correctly because the original equation has infinitely many solutions.