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.
Answers
GPT-4o mini
To determine if the student's steps to solve the equation \(4(2-3x)=x-2(2x+1)\) are correct, let's go through the solution step by step.
Starting with the original equation:
\[
4(2 - 3x) = x - 2(2x + 1)
\]
Step 1: Distribute on both sides.
Left side:
\[
4 \cdot 2 - 4 \cdot 3x = 8 - 12x
\]
Right side:
\[
x - 2(2x + 1) = x - 4x - 2 = x - 4x - 2 = -3x - 2
\]
Now, rewrite the equation with the distributed expressions:
\[
8 - 12x = -3x - 2
\]
Step 2: Move all terms involving \(x\) to one side and constant terms to the other side.
Add \(12x\) to both sides:
\[
8 = -3x + 12x - 2
\]
Simplifying this gives:
\[
8 = 9x - 2
\]
Step 3: Add \(2\) to both sides:
\[
8 + 2 = 9x
\]
This simplifies to:
\[
10 = 9x
\]
Step 4: Solve for \(x\):
\[
x = \frac{10}{9}
\]
Step 5: Conclusion
The student arrived at "No solution," which is incorrect. The correct solution is:
\[
x = \frac{10}{9}
\]
Thus, the student made a mistake in their calculations leading to the claim of no solution. The steps leading to their conclusion contain errors that should be corrected as shown above, leading to the valid solution instead.
Starting with the original equation:
\[
4(2 - 3x) = x - 2(2x + 1)
\]
Step 1: Distribute on both sides.
Left side:
\[
4 \cdot 2 - 4 \cdot 3x = 8 - 12x
\]
Right side:
\[
x - 2(2x + 1) = x - 4x - 2 = x - 4x - 2 = -3x - 2
\]
Now, rewrite the equation with the distributed expressions:
\[
8 - 12x = -3x - 2
\]
Step 2: Move all terms involving \(x\) to one side and constant terms to the other side.
Add \(12x\) to both sides:
\[
8 = -3x + 12x - 2
\]
Simplifying this gives:
\[
8 = 9x - 2
\]
Step 3: Add \(2\) to both sides:
\[
8 + 2 = 9x
\]
This simplifies to:
\[
10 = 9x
\]
Step 4: Solve for \(x\):
\[
x = \frac{10}{9}
\]
Step 5: Conclusion
The student arrived at "No solution," which is incorrect. The correct solution is:
\[
x = \frac{10}{9}
\]
Thus, the student made a mistake in their calculations leading to the claim of no solution. The steps leading to their conclusion contain errors that should be corrected as shown above, leading to the valid solution instead.