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

No solution.

(2 points)
Based on the student's work, the equation was solved

The equation solved correctly would show that it has
solution(s).

Answer 1

Let's analyze the problem step by step to verify if the student's solution is correct and to identify whether the original equation has a solution.

The original equation given is:

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

Expanding both sides:

1. Left Side:
\[ 4(2 - 3x) = 8 - 12x \]

2. Right Side:
\[ x - 2(2x + 1) = x - (4x + 2) = x - 4x - 2 = -3x - 2 \]

Now we have:

\[ 8 - 12x = -3x - 2 \]

Next, let's isolate \(x\). We can start by adding \(12x\) to both sides:

\[ 8 = 12x - 3x - 2 \]

This simplifies to:

\[ 8 = 9x - 2 \]

Now, add 2 to both sides:

\[ 10 = 9x \]

Dividing both sides by 9 gives:

\[ x = \frac{10}{9} \]

This means that the original equation does have a solution: \(x = \frac{10}{9}\).

Regarding the student's claim of "No solution," it appears they made an error in their calculations. Specifically, the step where they equated \(8 - 3x\) to \(-3x - 2\) was not followed through correctly in subsequent steps.

In conclusion, the original equation does have a solution:

**The equation solved correctly would show that it has 1 solution.**