`225x^(2)-4=0, sqrt225y +2=0`

Options

A. if `x gt y`
B. if `x gt= y`
C. if `x lt y`
D. if x= y or relation cannot be established

Answer

To solve the given equations \( 225x^2 - 4 = 0 \) and \( \sqrt{225y} + 2 = 0 \), we will find the values of \( x \) and \( y \) step by step.

### Step 1: Solve for \( x \)
We start with the equation:
\[ 225x^2 - 4 = 0 \]

**Hint:** Isolate the term with \( x^2 \).

Add \( 4 \) to both sides:
\[ 225x^2 = 4 \]

**Hint:** Divide both sides by \( 225 \) to solve for \( x^2 \).

Now divide by \( 225 \):
\[ x^2 = \frac{4}{225} \]

**Hint:** Take the square root of both sides to find \( x \).

Taking the square root gives:
\[ x = \pm \sqrt{\frac{4}{225}} \]
\[ x = \pm \frac{2}{15} \]

### Step 2: Solve for \( y \)
Next, we solve the equation:
\[ \sqrt{225y} + 2 = 0 \]

**Hint:** Isolate the square root term.

Subtract \( 2 \) from both sides:
\[ \sqrt{225y} = -2 \]

**Hint:** Consider the implications of a square root being negative.

Since the square root cannot be negative, there are no real solutions for \( y \) in this equation. Thus, we can conclude:
\[ y \text{ is undefined in real numbers.} \]

### Step 3: Compare \( x \) and \( y \)
Now we have:
- \( x = \pm \frac{2}{15} \)
- \( y \) is undefined.

**Hint:** Determine the relationship between \( x \) and \( y \).

Since \( y \) does not have a valid real number solution, we cannot establish a relationship between \( x \) and \( y \). Therefore, the correct conclusion is that the relation cannot be established.

### Final Answer
The values we found are:
- \( x = \pm \frac{2}{15} \)
- \( y \) is undefined.

Thus, the relation between \( x \) and \( y \) cannot be established.