Question

solve the equation below. whats the value of n? start by multiplying the left side of the equation by 2 to get rid of the fraction on the left side of the equation. \\(\frac{3}{2}(7 + 3n) = 3 + 3n\\) \\(2 \cdot \frac{3}{2}(7 + 3n) = 2 \cdot (3 + 3n)\\) \\(\square(7 + 3n) = ?\\)

Explanation:

Step1: Simplify Left Side

On the left side, \(2 \cdot \frac{3}{2}(7 + 3n)\), the 2 and \(\frac{1}{2}\) cancel out, leaving \(3(7 + 3n)\).

Step2: Simplify Right Side

On the right side, \(2 \cdot (3 + 3n)\) distributes to \(6 + 6n\).

Step3: Expand Left Side

Expand \(3(7 + 3n)\) to get \(21 + 9n\).

Step4: Solve for n

Set \(21 + 9n = 6 + 6n\). Subtract \(6n\) from both sides: \(21 + 3n = 6\). Subtract 21: \(3n = -15\). Divide by 3: \(n = -5\). But first, for the box and the right side simplification:
Left box: After \(2 \cdot \frac{3}{2}\), it's 3. Right side: \(2 \cdot (3 + 3n) = 6 + 6n\). Wait, the first part (box) is the coefficient after simplifying \(2 \cdot \frac{3}{2}\), which is 3. Then the right side is \(6 + 6n\). But let's follow the steps as per the problem's current step (the box and the?):

First, simplifying \(2 \cdot \frac{3}{2}(7 + 3n)\): the 2 and 2 in the denominator cancel, so it's \(3(7 + 3n)\), so the box is 3. Then the right side: \(2 \cdot (3 + 3n) = 6 + 6n\). But maybe the problem is just asking for the simplification of the left coefficient and the right side at that step. So the box is 3, and the? is \(6 + 6n\). But to find n:

After getting \(3(7 + 3n) = 2(3 + 3n)\) (wait, no, original after multiplying both sides: left is \(3(7 + 3n)\), right is \(2(3 + 3n)\)? Wait no, original equation after multiplying both sides: left is \(2 \cdot \frac{3}{2}(7 + 3n) = 3(7 + 3n)\), right is \(2(3 + 3n)\). Then expanding left: 21 + 9n, right: 6 + 6n. Then 21 + 9n = 6 + 6n → 3n = -15 → n = -5. But for the box in the left (the coefficient) is 3, and the right side is 6 + 6n. But let's do the first part (the box and the?):

Left side: \(2 \times \frac{3}{2} = 3\), so the box is 3. Right side: \(2 \times (3 + 3n) = 6 + 6n\), so the? is \(6 + 6n\). But maybe the problem is just asking for the coefficient in the box (3) and then later solving for n. Wait, the initial question is "What's the value of n?" but the image shows a step where we have to fill the box and the?. Let's solve the box first:

The left side: \(2 \cdot \frac{3}{2}(7 + 3n)\). The 2 and \(\frac{3}{2}\) multiply to \(3\) (since \(2 \times \frac{3}{2} = 3\)), so the box is 3. Then the right side: \(2 \cdot (3 + 3n) = 6 + 6n\), so the? is \(6 + 6n\). Then we can proceed to solve:

\(3(7 + 3n) = 6 + 6n\)

\(21 + 9n = 6 + 6n\)

\(9n - 6n = 6 - 21\)

\(3n = -15\)

\(n = -5\)

Answer:

The value of \(n\) is \(-5\). (For the box: 3; for the? on the right: \(6 + 6n\), but the final answer for \(n\) is \(-5\).)