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Sagot :
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].
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].
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