.Which expressions are equivalent to 12(1/2x - 3) + 12(1/3y + 2) + 6?

Question

Select all correct answers.
Responses
6(x+2/3y-1)

6x+4y-5

6x+4y-6

2(3x+2y-33)

2 ( 3 x + 2 y - 3 )

Answer 1

Let's simplify the expression \(12(1/2x - 3) + 12(1/3y + 2) + 6\).

1. Distributing the \(12\) in the first term: \[ 12(1/2x - 3) = 12 \cdot \frac{1}{2}x - 12 \cdot 3 = 6x - 36 \]

2. Distributing the \(12\) in the second term: \[ 12(1/3y + 2) = 12 \cdot \frac{1}{3}y + 12 \cdot 2 = 4y + 24 \]

3. Combine the results along with the \(6\): \[ 6x - 36 + 4y + 24 + 6 \]

4. Combine the constants: \[ -36 + 24 + 6 = -6 \]

5. So the entire expression simplifies to: \[ 6x + 4y - 6 \]

Now, let's check which of the given expressions are equivalent to \(6x + 4y - 6\).

Checking Each Option:

1. Option: \(6(x + \frac{2}{3}y - 1)\)

   • Distributing: \[ 6x + 4y - 6 \quad \text{(this matches)} \]

2. Option: \(6x + 4y - 5\)

   • This does not match \(6x + 4y - 6\).

3. Option: \(6x + 4y - 6\)

   • This matches \(6x + 4y - 6\).

4. Option: \(2(3x + 2y - 33)\)

   • Distributing: \[ 2 \cdot 3x + 2 \cdot 2y - 2 \cdot 33 = 6x + 4y - 66 \quad \text{(this does not match)} \]

5. Option: \(2(3x + 2y - 3)\)

   • Distributing: \[ 2 \cdot 3x + 2 \cdot 2y - 2 \cdot 3 = 6x + 4y - 6 \quad \text{(this matches)} \]

Summary of Equivalent Expressions:

• \(6(x + \frac{2}{3}y - 1)\)
• \(6x + 4y - 6\)
• \(2(3x + 2y - 3)\)

Therefore, the correct responses are:

• \(6(x + \frac{2}{3}y - 1)\)
• \(6x + 4y - 6\)
• \(2(3x + 2y - 3)\)