Let's carefully analyze the steps given to see if the isolation of \( x \) is performed correctly in the equation \( 3x + 2b = 6c \).
Starting with the original equation: \[ 3x + 2b = 6c \]
Step 1:
Subtract \( 2b \) from both sides: \[ 3x + 2b - 2b = 6c - 2b \] This simplifies to: \[ 3x = 6c - 2b \] This step is correct.
Step 2:
We have: \[ 3x = 6c - 2b \] This step is also correct.
Step 3:
Divide both sides by \( 3 \) to get \( x \) isolated: \[ x = \frac{6c - 2b}{3} \]
To simplify: \[ x = \frac{6c}{3} - \frac{2b}{3} \] This simplifies to: \[ x = 2c - \frac{2}{3}b \] This step is correct and gives us the final answer.
Final Answer:
So the final correct expression for \( x \) is: \[ x = 2c - \frac{2}{3}b \]
Evaluating Options:
Now, looking at the provided options:
- The statement that "the process is correct" is true but does not represent the final answer.
- The statement “The process has an error. The correct answer is \( x=2c−2b \)” is false.
- The statement “The process has an error. The correct answer is \( x=−4b−c \)” is false.
The option that states: “The process has an error. The correct answer is \( x \) is equal to \( 2c−\frac{2}{3}b \)” or its equivalent wording could be true, but it's not explicitly listed in your options.
Since the final answer accurately reflects the steps showing \( x \) to be \( 2c - \frac{2}{3}b \), and given the context, the conclusion is:
- "The process is correct" in isolation, but the final answer representation needs to state \( \frac{2}{3}b \) to be entirely accurate.
Conclusion:
The truthful statement should be that the process is correct, but it has an error in final presentation. The correct answer is: \[ x = 2c - \frac{2}{3}b \]