If `x^(2)` is added to `(5)/(4y)`, the sum is `(5+y)/(4y)`. If y is a positive integer, which of the following is value of x?

Options

A. `(1)/(4)`
B. `(1)/(2)`
C. `(4)/(5)`
D. `1`

Answer

To solve the problem step by step, we start with the equation given in the question.

### Step 1: Set up the equation
According to the problem, if \( x^2 \) is added to \( \frac{5}{4y} \), the sum is equal to \( \frac{5+y}{4y} \). We can write this as:
\[ x^2 + \frac{5}{4y} = \frac{5+y}{4y} \]

### Step 2: Isolate \( x^2 \)
Next, we want to isolate \( x^2 \) on one side of the equation. To do this, we can subtract \( \frac{5}{4y} \) from both sides:
\[ x^2 = \frac{5+y}{4y} - \frac{5}{4y} \]

### Step 3: Combine the fractions
Now, we can combine the fractions on the right side:
\[ x^2 = \frac{(5+y) - 5}{4y} = \frac{y}{4y} \]

### Step 4: Simplify the right side
We can simplify \( \frac{y}{4y} \):
\[ x^2 = \frac{1}{4} \]

### Step 5: Solve for \( x \)
Now, we take the square root of both sides to solve for \( x \):
\[ x = \pm \frac{1}{2} \]

### Step 6: Determine the value of \( x \)
Since the problem states that \( y \) is a positive integer, we can choose the positive value of \( x \):
\[ x = \frac{1}{2} \]

### Final Answer
Thus, the value of \( x \) is \( \frac{1}{2} \).

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