The given system of equations is:

1. \( 24x^2 + 9y^2 = 100 \)
2. \( -54y - 44 = 0 \)

### Step 1: Solve the second equation for \( y \).

The second equation is:
\[
-54y - 44 = 0
\]

Add 44 to both sides:
\[
-54y = 44
\]

Divide by -54:
\[
y = -\frac{44}{54} = -\frac{22}{27}
\]

So, \( y = -\frac{22}{27} \).

### Step 2: Substitute \( y = -\frac{22}{27} \) into the first equation.

The first equation is:
\[
24x^2 + 9y^2 = 100
\]

Substitute \( y = -\frac{22}{27} \):
\[
24x^2 + 9\left(-\frac{22}{27}\right)^2 = 100
\]

Simplify the square:
\[
24x^2 + 9\left(\frac{484}{729}\right) = 100
\]

Multiply the 9 by the fraction:
\[
24x^2 + \frac{4356}{729} = 100
\]

Simplify the fraction:
\[
24x^2 + \frac{6}{1} = 100
\]

Subtract 6 from both sides:
\[
24x^2 = 94
\]

Divide by 24:
\[
x^2 = \frac{94}{24} = \frac{47}{12}
\]

So:
\[
x = \pm \sqrt{\frac{47}{12}} = \pm \frac{\sqrt{47}}{2\sqrt{3}}
\]

### Final Answer:
The solution to the system is:
\[
x = \pm \frac{\sqrt{47}}{2\sqrt{3}}, \quad y = -\frac{22}{27}
\]

Would you like more details on this solution or have any questions?

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Here are five questions to consider:

1. How do you simplify complex fractions?
2. What are the properties of square roots?
3. How can you verify the solution of a system of equations?
4. What methods can be used to solve nonlinear equations?
5. How do you interpret solutions with radicals in practical applications?

**Tip:** When working with systems of equations, always try to simplify each equation as much as possible before solving the system.

This page provides a detailed step-by-step solution to the system of equations 24x^2 + 9y^2 = 100 and -54y - 44 = 0. Learn how to solve for x and y by applying algebraic techniques, including substitution and quadratic equation solutions. Suitable for students in Grades 10-12.

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