To solve the equation [tex]\(\frac{2x - 1}{y} = \frac{w + 2}{2z}\)[/tex] for [tex]\(w\)[/tex], follow these steps:
1. Start with the initial 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 the terms on each side:
[tex]\[
2z \cdot (2x - 1) = y \cdot (w + 2)
\][/tex]
This simplifies to:
[tex]\[
4xz - 2z = wy + 2y
\][/tex]
4. Isolate the term with [tex]\(w\)[/tex]:
[tex]\[
4xz - 2z - 2y = wy
\][/tex]
5. Subtract [tex]\(2y\)[/tex] on both sides to isolate [tex]\(wy\)[/tex]:
[tex]\[
4xz - 2z = wy + 2y \quad \Rightarrow \quad 4xz - 2z - 2y = wy
\][/tex]
6. Divide both sides by [tex]\(y\)[/tex] to solve for [tex]\(w\)[/tex]:
[tex]\[
w = \frac{4xz - 2z}{y} - 2
\][/tex]
Hence, the solution for [tex]\(w\)[/tex] in terms of [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex] is:
[tex]\[
w = \frac{4xz - 2z}{y}
\][/tex]
And this is the final simplified form of [tex]\(w\)[/tex].
