Question
3. 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).
Answers
Answered by
GPT-4o mini
Let's analyze the given equation and the student’s steps to see if the equation can be solved correctly and if a solution exists.
The original equation is:
\[
4(2 - 3x) = x - 2(2x + 1)
\]
Now, let's go through the student's steps:
1. Distributing on both sides:
- Left side: \(4(2 - 3x) = 8 - 12x\)
- Right side: \(x - 2(2x + 1) = x - 4x - 2 = -3x - 2\)
So, the equation becomes:
\[
8 - 12x = -3x - 2
\]
2. The student then rewrites it as:
\[
8 - 3x = -3x - 2
\]
This step appears to be incorrect. The correct step is to keep it as \(8 - 12x = -3x - 2\).
3. To isolate \(x\), we can add \(12x\) to both sides:
\[
8 = -3x + 12x - 2
\]
4. Simplifying this gives us:
\[
8 = 9x - 2
\]
5. Adding 2 to both sides:
\[
10 = 9x
\]
6. Dividing both sides by 9 gives:
\[
x = \frac{10}{9}
\]
Thus, the correct solution to the equation \(4(2-3x) = x - 2(2x+1)\) is \(x = \frac{10}{9}\).
The student's conclusion of "No solution" is incorrect. The equation does indeed have a solution, which is \(x = \frac{10}{9}\).
The original equation is:
\[
4(2 - 3x) = x - 2(2x + 1)
\]
Now, let's go through the student's steps:
1. Distributing on both sides:
- Left side: \(4(2 - 3x) = 8 - 12x\)
- Right side: \(x - 2(2x + 1) = x - 4x - 2 = -3x - 2\)
So, the equation becomes:
\[
8 - 12x = -3x - 2
\]
2. The student then rewrites it as:
\[
8 - 3x = -3x - 2
\]
This step appears to be incorrect. The correct step is to keep it as \(8 - 12x = -3x - 2\).
3. To isolate \(x\), we can add \(12x\) to both sides:
\[
8 = -3x + 12x - 2
\]
4. Simplifying this gives us:
\[
8 = 9x - 2
\]
5. Adding 2 to both sides:
\[
10 = 9x
\]
6. Dividing both sides by 9 gives:
\[
x = \frac{10}{9}
\]
Thus, the correct solution to the equation \(4(2-3x) = x - 2(2x+1)\) is \(x = \frac{10}{9}\).
The student's conclusion of "No solution" is incorrect. The equation does indeed have a solution, which is \(x = \frac{10}{9}\).