Find the best answers to your questions at Westonci.ca, where experts and enthusiasts provide accurate, reliable information. Join our Q&A platform and connect with professionals ready to provide precise answers to your questions in various areas. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
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.
Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.
