Question

solve the equation below. whats the value of z?
you can think of $2 \cdot \frac{3}{2}(4 + 3z)$ as $2 \cdot \frac{3}{2} \cdot (4 + 3z)$. you can multiply the first two factors first. what is $2 \cdot \frac{3}{2}$?
$\frac{3}{2}(4 + 3z) = 2 + 2z$
$2 \cdot \frac{3}{2}(4 + 3z) = 2 \cdot (2 + 2z)$
$\square (4 + 3z) = ?$

Explanation:

Step1: Calculate \(2 \cdot \frac{3}{2}\)

To find the product of \(2\) and \(\frac{3}{2}\), we multiply the numerator: \(2\times\frac{3}{2}=\frac{2\times3}{2}\). The \(2\) in the numerator and denominator cancels out, leaving \(3\).

Step2: Calculate \(2 \cdot (2 + 2z)\)

Using the distributive property \(a(b + c)=ab+ac\), where \(a = 2\), \(b = 2\), and \(c = 2z\). So, \(2\times2+2\times2z = 4 + 4z\).

Answer:

The value in the box is \(3\), and the right - hand side is \(4 + 4z\). So we have \(3(4 + 3z)=4 + 4z\) (for the first part of the boxed value and the right - hand side after multiplication). If we continue to solve for \(z\):

Step3: Expand the left - hand side

Using the distributive property \(3\times4+3\times3z=12 + 9z\). So the equation becomes \(12 + 9z=4 + 4z\).

Step4: Subtract \(4z\) from both sides

\(12 + 9z-4z=4 + 4z-4z\), which simplifies to \(12 + 5z=4\).

Step5: Subtract 12 from both sides

\(12 + 5z-12=4 - 12\), so \(5z=-8\).

Step6: Divide both sides by 5

\(z =-\frac{8}{5}=-1.6\)

If we only consider the first part of the question (finding \(2\cdot\frac{3}{2}\) and \(2\cdot(2 + 2z)\)):
The value in the box is \(3\) and the right - hand side is \(4 + 4z\). If we consider the final value of \(z\), \(z =-\frac{8}{5}\) (or \(-1.6\)).