To solve the equation [tex]\(\frac{2x - 1}{y} = \frac{w + 2}{2z}\)[/tex] for [tex]\(w\)[/tex], we will perform a systematic algebraic manipulation:
1. Start with the given equation:
[tex]\[
\frac{2x - 1}{y} = \frac{w + 2}{2z}
\][/tex]
2. Cross-multiply to eliminate the fractions:
[tex]\[
(2x - 1) \cdot 2z = (w + 2) \cdot y
\][/tex]
3. Distribute both sides:
[tex]\[
4xz - 2z = wy + 2y
\][/tex]
4. Rearrange to isolate the term involving [tex]\(w\)[/tex] on one side:
[tex]\[
wy = 4xz - 2z - 2y
\][/tex]
5. Solve for [tex]\(w\)[/tex] by dividing both sides by [tex]\(y\)[/tex]:
[tex]\[
w = \frac{4xz - 2z - 2y}{y}
\][/tex]
6. Let's simplify this expression by breaking it into separate fractions:
[tex]\[
w = \frac{4xz}{y} - \frac{2z}{y} - \frac{2y}{y}
\][/tex]
7. Simplify each term individually:
[tex]\[
w = \frac{4xz}{y} - \frac{2z}{y} - 2
\][/tex]
Thus, the simplified expression for [tex]\(w\)[/tex] is:
[tex]\[
w = \frac{4xz - 2z - 2y}{y}
\][/tex]
To further refine this, we notice:
[tex]\[
w = \frac{4z(x - 1) - 2y}{y}
\][/tex]
And if desired, it can be simplified using factoring more cleanly:
[tex]\[
w = \frac{4z(x - 1)}{y} - 2
\][/tex]
Comparing this result with the provided options:
- Option 1: [tex]\( w = \frac{4xz - 1}{y} - 2 \)[/tex] (incorrect, not matching)
- Option 2: [tex]\( w = \frac{4xz - 2z}{y} \)[/tex] (correct, before the [tex]\(-2y\)[/tex] adjustment, therefore not matching here)
- Option 3: [tex]\( w = \frac{xz - z}{y} - 2 \)[/tex] (incorrect, doesn't simplify correctly)
- Option 4: [tex]\( w = \frac{4xz - 2z}{y} - 2 \)[/tex] (correct, matches after the simplification)
Hence, the correct answer is:
[tex]\[
w = \frac{4xz - 2z}{y} - 2
\][/tex]
