Question

A student solved the following equation using the following steps:

4(2−3x)=x−2(2x+1)

8−3x=x−4x−2

8−3x=−3x−2

Answer: No Solution

Based on the student’s work, the equation was solved _____________

The equation solved correctly would show that it has _____________

Answer 1

Based on the student's work, the equation was solved **incorrectly**.

The equation solved correctly would show that it has **an infinite number of solutions (or that it is an identity)**, or it might show that it has **no solution** depending on how the variables simplify during the process.

To clarify: The student reached a conclusion of "No Solution," but we must check the steps on whether they led to a consistent or inconsistent equation. Upon closer inspection, the transition between equations 2 and 3 shows only the terms reorganized. Without seeing the exact steps of how the simplification occurred, it's not definitive that they concluded accurately.

Let's analyze the equation thoroughly from the beginning to determine the actual outcome.

Starting from:
\[ 4(2 - 3x) = x - 2(2x + 1) \]

First, we distribute:
\[ 8 - 12x = x - (4x + 2) \]

This simplifies to:
\[ 8 - 12x = x - 4x - 2 \]
\[ 8 - 12x = -3x - 2 \]

Rearranging gives:
\[ 8 + 2 = -3x + 12x \]
\[ 10 = 9x \]

From here, we isolate x:
\[ x = \frac{10}{9} \]

This indicates that there is indeed a solution, as the value can be calculated. Therefore, the correct completion would be:

- Based on the student's work, the equation was solved **incorrectly**.
- The equation solved correctly would show that it has **one unique solution**.

Hence, it is vital to ensure the mathematical processes stay accurate throughout.