Question

solve the equation below. whats the value of z? 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}(4 + 3z) = 2 + 2z\\) \\(2 \cdot \frac{3}{2}(4 + 3z) = 2 \cdot (2 + 2z)\\) \\(\square(4 + 3z) = ?\\)

Explanation:

Step1: Simplify left side

Multiply \(2\) and \(\frac{3}{2}\): \(2\cdot\frac{3}{2} = 3\), so left side becomes \(3(4 + 3z)\).

Step2: Simplify right side

Multiply \(2\) and \((2 + 2z)\): \(2\cdot(2 + 2z)=4 + 4z\).

Step3: Expand left side

\(3(4 + 3z)=12 + 9z\).

Step4: Solve for \(z\)

Set \(12 + 9z = 4 + 4z\). Subtract \(4z\) and \(12\): \(9z - 4z = 4 - 12\) → \(5z=-8\) → \(z = -\frac{8}{5}\). But first, for the box and "?": After step1 and step2, left box is \(3\), right "?" is \(4 + 4z\). Wait, the question's first part: after multiplying by 2, left side \(2\cdot\frac{3}{2}(4 + 3z)=3(4 + 3z)\), so box is \(3\); right side \(2\cdot(2 + 2z)=4 + 4z\), so "?" is \(4 + 4z\). Then solving fully:
From \(3(4 + 3z)=4 + 4z\), expand: \(12 + 9z = 4 + 4z\), subtract \(4z\): \(12 + 5z = 4\), subtract \(12\): \(5z=-8\), \(z = -\frac{8}{5}\) or \(-1.6\). But the initial steps: first, the box (coefficient) is \(3\), right side after multiplying by 2 is \(4 + 4z\). Then solving for \(z\):

Wait, the problem's first part: "Start by multiplying...", so first, left side: \(2\cdot\frac{3}{2}=3\), so box is \(3\); right side: \(2\cdot(2 + 2z)=4 + 4z\), so "?" is \(4 + 4z\). Then proceed to solve:

\(3(4 + 3z)=4 + 4z\)

\(12 + 9z = 4 + 4z\)

\(9z - 4z = 4 - 12\)

\(5z = -8\)

\(z = -\frac{8}{5}\)

But let's follow the steps as per the problem's instruction (first, fill the box and "?", then solve for \(z\)).

First, the box: \(2\cdot\frac{3}{2}=3\), so box is \(3\).

The "?": \(2\cdot(2 + 2z)=4 + 4z\), so "?" is \(4 + 4z\).

Then, solving the equation \(3(4 + 3z)=4 + 4z\):

Step1 (box):

Calculate \(2\times\frac{3}{2}\): \(2\times\frac{3}{2}=3\), so box is \(3\).

Step2 (right side after multiply):

Calculate \(2\times(2 + 2z)\): \(2\times2 + 2\times2z = 4 + 4z\), so "?" is \(4 + 4z\).

Step3 (expand left):

\(3(4 + 3z)=12 + 9z\).

Step4 (set equal and solve):

\(12 + 9z = 4 + 4z\)

Subtract \(4z\) from both sides: \(12 + 5z = 4\)

Subtract \(12\) from both sides: \(5z = -8\)

Divide by \(5\): \(z = -\frac{8}{5}\) (or \(-1.6\))

Answer:

First, the box value is \(3\), the "?" value is \(4 + 4z\), and the value of \(z\) is \(-\frac{8}{5}\) (or \(-1.6\)). But if the question is asking for \(z\) after solving, then the final answer for \(z\) is \(\boxed{-\dfrac{8}{5}}\) (or \(\boxed{-1.6}\)).