Question
When solving the problem 45x−3=16x+6 , your friend does the following: 45x−3=4x+6 5x−3=x+6 4x=9 x=94 Was this correct? (1 point) Responses No, in step 1 when the 16 changed to 4, it should have been 42 and the 2 distributed to the current exponent. No, in step 1 when the 16 changed to 4, it should have been 4 squared and the 2 distributed to the current exponent. No, the bases of 4 and 16 should have both been changed to a base of 2. No, the bases of 4 and 16 should have both been changed to a base of 2. Yes, this is correct. No error was made. Yes, this is correct. No error was made. No, in step 2, the 4 should have been distributed to the current exponents. This would change the equation and yield a different final answer.
Answers
GPT-4o mini
To solve the equation \( 45x - 3 = 16x + 6 \), let's go through your friend's steps as you've presented them:
1. **Starting Equation**:
\[
45x - 3 = 16x + 6
\]
2. **Step 1**: Your friend incorrectly wrote:
\[
45x - 3 = 4x + 6
\]
Here, the term \( 16x \) incorrectly changed to \( 4x \). The correct form of the equation should still have \( 16x \).
3. **Step 2**: Your friend then simplified it to:
\[
5x - 3 = x + 6
\]
This step is a continuation of the error, as \( 4x \) was used instead of \( 16x \).
4. **Step 3**: He further simplified:
\[
4x = 9
\]
This step does not follow correctly from the previous equation.
5. **Final Result**: Your friend concluded with:
\[
x = 94
\]
This is also incorrect due to the earlier mistakes.
Now, reviewing the options provided:
1. **No, in step 1 when the 16 changed to 4, it should have been 42 and the 2 distributed to the current exponent.** (Not relevant)
2. **No, in step 1 when the 16 changed to 4, it should have been 4 squared and the 2 distributed to the current exponent.** (Not relevant)
3. **No, the bases of 4 and 16 should have both been changed to a base of 2.** (Correct, but not applicable here since the error was at the direct number level)
4. **No, the bases of 4 and 16 should have both been changed to a base of 2.** (Repeated option)
5. **Yes, this is correct. No error was made.** (Incorrect, as errors were made)
6. **No, in step 2, the 4 should have been distributed to the current exponents. This would change the equation and yield a different final answer.** (Also not relevant as step 2 follows an incorrect step 1)
The most correct interpretation is that **the major error occurred in step 1**, where \( 16x \) was incorrectly changed to \( 4x \). Thus, the first response option is the most relevant, noting that there are issues with bases, but the immediate transaction of numbers was the key error. Therefore, the answer regarding the correctness of your friend's approach is **No, in step 1 when the 16 changed to 4, it should have been 16.**
1. **Starting Equation**:
\[
45x - 3 = 16x + 6
\]
2. **Step 1**: Your friend incorrectly wrote:
\[
45x - 3 = 4x + 6
\]
Here, the term \( 16x \) incorrectly changed to \( 4x \). The correct form of the equation should still have \( 16x \).
3. **Step 2**: Your friend then simplified it to:
\[
5x - 3 = x + 6
\]
This step is a continuation of the error, as \( 4x \) was used instead of \( 16x \).
4. **Step 3**: He further simplified:
\[
4x = 9
\]
This step does not follow correctly from the previous equation.
5. **Final Result**: Your friend concluded with:
\[
x = 94
\]
This is also incorrect due to the earlier mistakes.
Now, reviewing the options provided:
1. **No, in step 1 when the 16 changed to 4, it should have been 42 and the 2 distributed to the current exponent.** (Not relevant)
2. **No, in step 1 when the 16 changed to 4, it should have been 4 squared and the 2 distributed to the current exponent.** (Not relevant)
3. **No, the bases of 4 and 16 should have both been changed to a base of 2.** (Correct, but not applicable here since the error was at the direct number level)
4. **No, the bases of 4 and 16 should have both been changed to a base of 2.** (Repeated option)
5. **Yes, this is correct. No error was made.** (Incorrect, as errors were made)
6. **No, in step 2, the 4 should have been distributed to the current exponents. This would change the equation and yield a different final answer.** (Also not relevant as step 2 follows an incorrect step 1)
The most correct interpretation is that **the major error occurred in step 1**, where \( 16x \) was incorrectly changed to \( 4x \). Thus, the first response option is the most relevant, noting that there are issues with bases, but the immediate transaction of numbers was the key error. Therefore, the answer regarding the correctness of your friend's approach is **No, in step 1 when the 16 changed to 4, it should have been 16.**