I. ` 10x^(2) - 7x + 1= 0" "`
II. ` 35y^(2) - 12y + 1 = 0 `

Options

A. if ` x lt y`
B. if ` x gt y`
C. if ` x = y`
D. if `x ge y`

Explanation

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

### Step 1: Solve the first equation \(10x^2 - 7x + 1 = 0\)

1. **Identify the coefficients**:
- \(a = 10\), \(b = -7\), \(c = 1\)

2. **Use the quadratic formula**:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

3. **Calculate the discriminant**:
\[ b^2 - 4ac = (-7)^2 - 4 \cdot 10 \cdot 1 = 49 - 40 = 9 \]

4. **Find the roots**:
\[ x = \frac{7 \pm \sqrt{9}}{20} = \frac{7 \pm 3}{20} \]
- First root:
\[ x_1 = \frac{7 + 3}{20} = \frac{10}{20} = \frac{1}{2} \]
- Second root:
\[ x_2 = \frac{7 - 3}{20} = \frac{4}{20} = \frac{1}{5} \]

### Step 2: Solve the second equation \(35y^2 - 12y + 1 = 0\)

1. **Identify the coefficients**:
- \(a = 35\), \(b = -12\), \(c = 1\)

2. **Use the quadratic formula**:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

3. **Calculate the discriminant**:
\[ b^2 - 4ac = (-12)^2 - 4 \cdot 35 \cdot 1 = 144 - 140 = 4 \]

4. **Find the roots**:
\[ y = \frac{12 \pm \sqrt{4}}{70} = \frac{12 \pm 2}{70} \]
- First root:
\[ y_1 = \frac{12 + 2}{70} = \frac{14}{70} = \frac{1}{5} \]
- Second root:
\[ y_2 = \frac{12 - 2}{70} = \frac{10}{70} = \frac{1}{7} \]

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

1. **Values of \(x\)**:
- \(x_1 = \frac{1}{2}\)
- \(x_2 = \frac{1}{5}\)

2. **Values of \(y\)**:
- \(y_1 = \frac{1}{5}\)
- \(y_2 = \frac{1}{7}\)

3. **Comparison**:
- Comparing \(x_1\) and \(y_1\):
\[ \frac{1}{2} > \frac{1}{5} \]
- Comparing \(x_2\) and \(y_1\):
\[ \frac{1}{5} = \frac{1}{5} \]
- Comparing \(x_1\) and \(y_2\):
\[ \frac{1}{2} > \frac{1}{7} \]

### Conclusion:
From the comparisons, we can conclude that:
\[ x \geq y \]

### Final Answer:
Thus, the relation between \(x\) and \(y\) is \(x \geq y\).

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