In each question two equations numbered I and II are given. You have to salve both the equations and mark the answer
I. `16x^2-14x+3=0`
II. `6y^2-19y+15=0`

Options

A. `x gt y`
B. `x ge y `
C. `x lt y `
D. `x le y `

Answer

To solve the given equations, we will follow these steps:

### Step 1: Solve the first equation \( I: 16x^2 - 14x + 3 = 0 \)

1. **Identify the coefficients**:
- \( a = 16 \)
- \( b = -14 \)
- \( c = 3 \)

2. **Factor the quadratic equation**:
- We need to express \(-14x\) as a combination of two terms that multiply to \(16 \times 3 = 48\).
- We can split \(-14x\) into \(-8x\) and \(-6x\):
\[ 16x^2 - 8x - 6x + 3 = 0 \]

3. **Group the terms**:
\[ (16x^2 - 8x) + (-6x + 3) = 0 \]

4. **Factor by grouping**:
- From the first group, factor out \(8x\):
\[ 8x(2x - 1) - 3(2x - 1) = 0 \]
- Now factor out \((2x - 1)\):
\[ (2x - 1)(8x - 3) = 0 \]

5. **Set each factor to zero**:
- \(2x - 1 = 0 \Rightarrow x = \frac{1}{2}\)
- \(8x - 3 = 0 \Rightarrow x = \frac{3}{8}\)

### Step 2: Solve the second equation \( II: 6y^2 - 19y + 15 = 0 \)

1. **Identify the coefficients**:
- \( a = 6 \)
- \( b = -19 \)
- \( c = 15 \)

2. **Factor the quadratic equation**:
- We need to express \(-19y\) as a combination of two terms that multiply to \(6 \times 15 = 90\).
- We can split \(-19y\) into \(-9y\) and \(-10y\):
\[ 6y^2 - 9y - 10y + 15 = 0 \]

3. **Group the terms**:
\[ (6y^2 - 9y) + (-10y + 15) = 0 \]

4. **Factor by grouping**:
- From the first group, factor out \(3y\):
\[ 3y(2y - 3) - 5(2y - 3) = 0 \]
- Now factor out \((2y - 3)\):
\[ (2y - 3)(3y - 5) = 0 \]

5. **Set each factor to zero**:
- \(2y - 3 = 0 \Rightarrow y = \frac{3}{2}\)
- \(3y - 5 = 0 \Rightarrow y = \frac{5}{3}\)

### Step 3: Compare the values of \(x\) and \(y\)

- The values obtained are:
- \(x = \frac{1}{2} \) and \(x = \frac{3}{8}\)
- \(y = \frac{3}{2} \) and \(y = \frac{5}{3}\)

### Step 4: Determine the relationship between \(x\) and \(y\)

- Since both values of \(y\) (\(\frac{3}{2}\) and \(\frac{5}{3}\)) are greater than both values of \(x\) (\(\frac{1}{2}\) and \(\frac{3}{8}\)), we conclude that \(y > x\).

### Final Answer
The correct answer is \( x < y \).

---