(i) ` x^(2) - 4x + 3 = 0`
(ii) ` y^(2) - 7y+10 = 0`

Options

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

Explanation

To solve the equations given in the question step by step, we will first solve each quadratic equation separately and then analyze the relationship between the values of \(x\) and \(y\).

### Step 1: Solve the first equation \(x^2 - 4x + 3 = 0\)

1. **Identify the equation**: \(x^2 - 4x + 3 = 0\)
2. **Factor the quadratic**: We need to find two numbers that multiply to \(3\) (the constant term) and add up to \(-4\) (the coefficient of \(x\)). The numbers are \(-3\) and \(-1\).
\[ x^2 - 3x - 1x + 3 = 0 \]
\[ (x - 3)(x - 1) = 0 \]
3. **Set each factor to zero**:
\[ x - 3 = 0 \quad \text{or} \quad x - 1 = 0 \]
4. **Solve for \(x\)**:
\[ x = 3 \quad \text{or} \quad x = 1 \]

### Step 2: Solve the second equation \(y^2 - 7y + 10 = 0\)

1. **Identify the equation**: \(y^2 - 7y + 10 = 0\)
2. **Factor the quadratic**: We need to find two numbers that multiply to \(10\) (the constant term) and add up to \(-7\) (the coefficient of \(y\)). The numbers are \(-5\) and \(-2\).
\[ y^2 - 5y - 2y + 10 = 0 \]
\[ (y - 5)(y - 2) = 0 \]
3. **Set each factor to zero**:
\[ y - 5 = 0 \quad \text{or} \quad y - 2 = 0 \]
4. **Solve for \(y\)**:
\[ y = 5 \quad \text{or} \quad y = 2 \]

### Step 3: Analyze the relationship between \(x\) and \(y\)

Now we have the values for \(x\) and \(y\):
- \(x\) can be \(1\) or \(3\)
- \(y\) can be \(2\) or \(5\)

We will analyze the possible relationships:

1. **If \(x = 1\)**:
- \(1 < 2\) (True)
- \(1 < 5\) (True)

2. **If \(x = 3\)**:
- \(3 > 2\) (True)
- \(3 > 5\) (False)

From this analysis, we see that:
- When \(x = 1\), it is less than both values of \(y\).
- When \(x = 3\), it is greater than \(2\) but not greater than \(5\).

### Conclusion

Since there is no consistent relationship that holds for both values of \(x\) and \(y\), we conclude that there is **no relation** between \(x\) and \(y\).

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