Answer :
To solve the equation [tex]\(\frac{2x - 1}{y} = \frac{w + 2}{2z}\)[/tex] for [tex]\(w\)[/tex], let's go through each step in detail:
### Step-by-Step Solution:
1. Given Equation:
[tex]\[ \frac{2x - 1}{y} = \frac{w + 2}{2z} \][/tex]
2. Cross-Multiply to Eliminate the Fractions:
[tex]\[ (2x - 1) \cdot (2z) = y \cdot (w + 2) \][/tex]
3. Distribute the Terms:
[tex]\[ 2z(2x - 1) = y(w + 2) \\ 4xz - 2z = y(w + 2) \][/tex]
4. Isolate the Term with [tex]\(w\)[/tex]:
[tex]\[ 4xz - 2z = yw + 2y \][/tex]
5. Solve for [tex]\(w\)[/tex]:
- Subtract [tex]\(2y\)[/tex] from both sides to isolate [tex]\(yw\)[/tex]:
[tex]\[ 4xz - 2z - 2y = yw \][/tex]
- Divide both sides by [tex]\(y\)[/tex] to solve for [tex]\(w\)[/tex]:
[tex]\[ w = \frac{4xz - 2z - 2y}{y} \][/tex]
6. Alternatively, Simplify the Expression :
[tex]\[ w = \frac{4xz - 2(y + z)}{y} \][/tex]
### Checking Possible Solutions:
- Given possible solutions:
[tex]\[ \frac{4xz - 1}{y} - 2, \quad \frac{4xz - 2z}{y}, \quad \frac{xz - z}{y} - 2, \quad \frac{4xz - 2z}{y} - 2 \][/tex]
7. Evaluate Each Possible Solution:
- First Possible Solution:
[tex]\[ w = \frac{4xz - 1}{y} - 2 \][/tex]
- Second Possible Solution:
[tex]\[ w = \frac{4xz - 2z}{y} \][/tex]
- Third Possible Solution:
[tex]\[ w = \frac{xz - z}{y} - 2 \][/tex]
- Fourth Possible Solution:
[tex]\[ w = \frac{4xz - 2z}{y} - 2 \][/tex]
### Conclusion:
- Comparing the derived solution [tex]\(w = \frac{4xz - 2(y + z)}{y}\)[/tex] with the given options:
The correct answer for [tex]\(w\)[/tex] based on the algebraic manipulation is:
[tex]\[ w = 2\left(\frac{2xz - y - z}{y}\right) \][/tex]
- This matches with the correct form:
[tex]\[ w = 2\left(\frac{4xz - 2z - 2y}{y}\right) \][/tex]
And, thus, the detailed solution aligns well with the possible given solutions. The list matches with the fact we're simplifying and expressing the alternatives.
### Step-by-Step Solution:
1. Given Equation:
[tex]\[ \frac{2x - 1}{y} = \frac{w + 2}{2z} \][/tex]
2. Cross-Multiply to Eliminate the Fractions:
[tex]\[ (2x - 1) \cdot (2z) = y \cdot (w + 2) \][/tex]
3. Distribute the Terms:
[tex]\[ 2z(2x - 1) = y(w + 2) \\ 4xz - 2z = y(w + 2) \][/tex]
4. Isolate the Term with [tex]\(w\)[/tex]:
[tex]\[ 4xz - 2z = yw + 2y \][/tex]
5. Solve for [tex]\(w\)[/tex]:
- Subtract [tex]\(2y\)[/tex] from both sides to isolate [tex]\(yw\)[/tex]:
[tex]\[ 4xz - 2z - 2y = yw \][/tex]
- Divide both sides by [tex]\(y\)[/tex] to solve for [tex]\(w\)[/tex]:
[tex]\[ w = \frac{4xz - 2z - 2y}{y} \][/tex]
6. Alternatively, Simplify the Expression :
[tex]\[ w = \frac{4xz - 2(y + z)}{y} \][/tex]
### Checking Possible Solutions:
- Given possible solutions:
[tex]\[ \frac{4xz - 1}{y} - 2, \quad \frac{4xz - 2z}{y}, \quad \frac{xz - z}{y} - 2, \quad \frac{4xz - 2z}{y} - 2 \][/tex]
7. Evaluate Each Possible Solution:
- First Possible Solution:
[tex]\[ w = \frac{4xz - 1}{y} - 2 \][/tex]
- Second Possible Solution:
[tex]\[ w = \frac{4xz - 2z}{y} \][/tex]
- Third Possible Solution:
[tex]\[ w = \frac{xz - z}{y} - 2 \][/tex]
- Fourth Possible Solution:
[tex]\[ w = \frac{4xz - 2z}{y} - 2 \][/tex]
### Conclusion:
- Comparing the derived solution [tex]\(w = \frac{4xz - 2(y + z)}{y}\)[/tex] with the given options:
The correct answer for [tex]\(w\)[/tex] based on the algebraic manipulation is:
[tex]\[ w = 2\left(\frac{2xz - y - z}{y}\right) \][/tex]
- This matches with the correct form:
[tex]\[ w = 2\left(\frac{4xz - 2z - 2y}{y}\right) \][/tex]
And, thus, the detailed solution aligns well with the possible given solutions. The list matches with the fact we're simplifying and expressing the alternatives.
