Let's solve the equations and evaluate the responses one by one:
Question 1
Solve the equation for y:
7y = 12x
To isolate y, divide both sides by 7:
\[ y = \frac{12x}{7} \]
This gives us:
\[ y = \frac{12}{7}x \] or equivalently \( y = \frac{12}{7}x \)
Correct Response: \[ y = \frac{12}{7}x \text{ is equal to } \frac{12}{7}x \]
Question 2
Change the equation from standard form to slope-intercept form:
\[ 8x - 4y = 24 \]
Bob's Steps:
- Subtract \( 8x \): \(-4y = -8x + 24\)
- Divide by \(-4\): \( y = 2x - 6 \) (This step is correct.)
Bill's Steps:
- Subtract \( 8x \): \(-4y = 16x\) (Incorrect because it should be \(-8x\).)
Bud's Steps:
- Subtract \( 8x \): \(-4y = -8x + 24\) (Correct)
- Divide by \(-4\): \( y = 2x + 6 \) (Incorrect result.)
Based on the evaluation, the correct response is: Bob has the correct answer.
Question 3
Solve the equation for x:
\[ 3x + 2b = 6c \]
Steps:
- Subtract \( 2b \) from both sides:
\[ 3x = 6c - 2b \] - Divide by 3 on both sides:
\[ x = \frac{6c - 2b}{3} \]
Which simplifies to:
\[ x = 2c - \frac{2}{3}b \]
The given process is:
Step 4: \( x = 2c - 2b \) (This part is incorrect; it should have been \( x = 2c - \frac{2}{3}b \).)
Correct Response:
The process has an error. The correct answer is \( x = 2c - \frac{2}{3}b \).
Final Responses
- Question 1: \( y = \frac{12}{7}x \)
- Question 2: Bob
- Question 3: The process has an error. The correct answer is \( x = 2c - \frac{2}{3}b \)