3𝑥^1/2-y^-1/2=4       and 4𝑥^1/2+3y^-1/2=14

We are given the system of two equations:

\[
1. \  3x^{1/2} - y^{-1/2} = 4
\]
\[
2. \  4x^{1/2} + 3y^{-1/2} = 14
\]

### Steps to Solve:

1. Let \( a = x^{1/2} \) and \( b = y^{-1/2} \).
   - This simplifies the equations to:
     \[
     1. \  3a - b = 4
     \]
     \[
     2. \  4a + 3b = 14
     \]

2. Solve the first equation for \( b \):
   \[
   b = 3a - 4
   \]

3. Substitute this expression for \( b \) into the second equation:
   \[
   4a + 3(3a - 4) = 14
   \]
   Simplify:
   \[
   4a + 9a - 12 = 14
   \]
   Combine like terms:
   \[
   13a - 12 = 14
   \]
   Add 12 to both sides:
   \[
   13a = 26
   \]
   Solve for \( a \):
   \[
   a = 2
   \]

4. Substitute \( a = 2 \) back into the expression for \( b \):
   \[
   b = 3(2) - 4 = 6 - 4 = 2
   \]

5. Now, recall that \( a = x^{1/2} \) and \( b = y^{-1/2} \). Thus:
   \[
   x^{1/2} = 2 \implies x = 4
   \]
   \[
   y^{-1/2} = 2 \implies y^{1/2} = \frac{1}{2} \implies y = \frac{1}{4}
   \]

### Final Answer:
\[
x = 4, \ y = \frac{1}{4}
\]

Would you like further details or have any questions?

Here are 5 related questions to explore:
1. What are the properties of square roots and their inverses?
2. How can we solve systems of equations using substitution?
3. How does changing the exponent affect the behavior of variables in an equation?
4. What is the geometric interpretation of equations with square roots?
5. How do negative exponents relate to inverse functions?

**Tip:** When simplifying complex equations, substituting variables often makes solving much easier!

3x^1/2 - y^-1/2 = 4 and 4x^1/2 + 3y^-1/2 = 14

Explore how to solve the system of equations 3x^1/2 - y^-1/2 = 4 and 4x^1/2 + 3y^-1/2 = 14 using substitution. Learn key algebraic concepts like square roots, inverse exponents, and solving systems of equations. Suitable for high school students, Grades 9-12.

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