Question

solve the equation.
4(3z + 4) = 4 - 3(z - 4)
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. z = \boxed{} (type an integer or a simplified fraction.)
b. the solution is all real numbers.
c. there is no solution.

Explanation:

Step1: Expand both sides

Using the distributive property \(a(b + c)=ab+ac\), we expand the left - hand side \(4(3z + 4)\) to get \(4\times3z+4\times4 = 12z+16\), and the right - hand side \(4-3(z - 4)\) to get \(4-3z + 12=16-3z\). So the equation becomes \(12z + 16=16-3z\).

Step2: Add \(3z\) to both sides

Adding \(3z\) to both sides of the equation \(12z + 16=16-3z\), we have \(12z+3z + 16=16-3z+3z\), which simplifies to \(15z+16 = 16\).

Step3: Subtract 16 from both sides

Subtracting 16 from both sides of the equation \(15z+16 = 16\), we get \(15z+16 - 16=16 - 16\), so \(15z=0\).

Step4: Divide both sides by 15

Dividing both sides of the equation \(15z = 0\) by 15, we have \(\frac{15z}{15}=\frac{0}{15}\), which gives \(z = 0\).

Answer:

A. \(z = 0\)