In the following questions, two equations numberded I and II are given. You have to solve both the equations and give answer If:
`I. 22x^2 -x-21=0 II. 22y^2 +43y+21=0`

Options

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

Answer

To solve the given equations step by step, we will first tackle each equation separately.

### Step 1: Solve Equation I
The first equation is:
\[ 22x^2 - x - 21 = 0 \]

To solve this quadratic equation, we can use the factorization method. We need to find two numbers that multiply to \( 22 \times (-21) = -462 \) and add up to \(-1\) (the coefficient of \(x\)).

#### Finding the Factors
The factors of \(-462\) that add up to \(-1\) are:
- \(21\) and \(-22\)

#### Rewriting the Equation
We can rewrite the middle term using these factors:
\[ 22x^2 + 21x - 22x - 21 = 0 \]

#### Grouping
Now, group the terms:
\[ (22x^2 + 21x) + (-22x - 21) = 0 \]

Factor by grouping:
\[ 7x(3x + 1) - 21(3x + 1) = 0 \]

This gives:
\[ (3x + 1)(22x - 21) = 0 \]

#### Finding the Roots
Setting each factor to zero:
1. \(3x + 1 = 0 \Rightarrow x = -\frac{1}{3}\)
2. \(22x - 21 = 0 \Rightarrow x = \frac{21}{22}\)

### Step 2: Solve Equation II
The second equation is:
\[ 22y^2 + 43y + 21 = 0 \]

Again, we will use the factorization method. We need to find two numbers that multiply to \(22 \times 21 = 462\) and add up to \(43\).

#### Finding the Factors
The factors of \(462\) that add up to \(43\) are:
- \(21\) and \(22\)

#### Rewriting the Equation
We can rewrite the middle term using these factors:
\[ 22y^2 + 21y + 22y + 21 = 0 \]

#### Grouping
Now, group the terms:
\[ (22y^2 + 21y) + (22y + 21) = 0 \]

Factor by grouping:
\[ y(22y + 21) + 1(22y + 21) = 0 \]

This gives:
\[ (22y + 21)(y + 1) = 0 \]

#### Finding the Roots
Setting each factor to zero:
1. \(22y + 21 = 0 \Rightarrow y = -\frac{21}{22}\)
2. \(y + 1 = 0 \Rightarrow y = -1\)

### Step 3: Compare Values of x and y
Now we have the values:
- From Equation I: \(x = -\frac{1}{3}, \frac{21}{22}\)
- From Equation II: \(y = -\frac{21}{22}, -1\)

#### Comparing Values
1. **For \(x = -\frac{1}{3}\) and \(y = -\frac{21}{22}\)**:
- \(-\frac{1}{3} \approx -0.33\) and \(-\frac{21}{22} \approx -0.95\)
- Here, \(x > y\).

2. **For \(x = -\frac{1}{3}\) and \(y = -1\)**:
- \(-\frac{1}{3} \approx -0.33\) and \(-1\)
- Here, \(x > y\).

3. **For \(x = \frac{21}{22}\) and \(y = -\frac{21}{22}\)**:
- \(\frac{21}{22} \approx 0.95\) and \(-\frac{21}{22} \approx -0.95\)
- Here, \(x > y\).

4. **For \(x = \frac{21}{22}\) and \(y = -1\)**:
- \(\frac{21}{22} \approx 0.95\) and \(-1\)
- Here, \(x > y\).

### Conclusion
In all cases, we find that \(x\) is greater than \(y\).

### Final Answer
Thus, the relation between \(x\) and \(y\) is:
\[ x > y \]