Two equations (I) and (II) are given in each question. On the basis of these equations you have to decide the relation between x and y and give answer
I. ` 5 x^(2) + 3x - 14 = 0 `
II. ` 10 y^(2) - 3 y - 27 = 0 `

Options

A. If ` x gt y `
B. If ` x lt y `
C. If ` x ge Y`
D. If x = y or no relation can be established between 'x' and 'y'

Answer

To solve the problem, we will analyze the two equations provided, find the values of \( x \) and \( y \), and then determine the relationship between them.

### Step 1: Solve Equation I
The first equation is:
\[ 5x^2 + 3x - 14 = 0 \]

To solve this quadratic equation, we will use the factorization method. We need to find two numbers that multiply to \( 5 \times (-14) = -70 \) and add up to \( 3 \).

The numbers that satisfy these conditions are \( 10 \) and \( -7 \). Therefore, we can rewrite the equation as:
\[ 5x^2 + 10x - 7x - 14 = 0 \]

Now, we can group the terms:
\[ (5x^2 + 10x) + (-7x - 14) = 0 \]

Factoring out the common terms:
\[ 5x(x + 2) - 7(x + 2) = 0 \]

Now, we can factor by grouping:
\[ (5x - 7)(x + 2) = 0 \]

Setting each factor to zero gives us:
1. \( 5x - 7 = 0 \) ⇒ \( x = \frac{7}{5} = 1.4 \)
2. \( x + 2 = 0 \) ⇒ \( x = -2 \)

### Step 2: Solve Equation II
The second equation is:
\[ 10y^2 - 3y - 27 = 0 \]

Again, we will use the factorization method. We need to find two numbers that multiply to \( 10 \times (-27) = -270 \) and add up to \( -3 \).

The numbers that satisfy these conditions are \( 15 \) and \( -18 \). Therefore, we can rewrite the equation as:
\[ 10y^2 + 15y - 18y - 27 = 0 \]

Now, we can group the terms:
\[ (10y^2 + 15y) + (-18y - 27) = 0 \]

Factoring out the common terms:
\[ 5y(2y + 3) - 9(2y + 3) = 0 \]

Now, we can factor by grouping:
\[ (5y - 9)(2y + 3) = 0 \]

Setting each factor to zero gives us:
1. \( 5y - 9 = 0 \) ⇒ \( y = \frac{9}{5} = 1.8 \)
2. \( 2y + 3 = 0 \) ⇒ \( y = -\frac{3}{2} = -1.5 \)

### Step 3: Compare Values of \( x \) and \( y \)
Now we have the values:
- From Equation I: \( x = 1.4 \) and \( x = -2 \)
- From Equation II: \( y = 1.8 \) and \( y = -1.5 \)

Now, we will compare the values of \( x \) and \( y \):

1. For \( x = 1.4 \):
- Compare with \( y = 1.8 \): \( 1.4 < 1.8 \)
- Compare with \( y = -1.5 \): \( 1.4 > -1.5 \)

2. For \( x = -2 \):
- Compare with \( y = 1.8 \): \( -2 < 1.8 \)
- Compare with \( y = -1.5 \): \( -2 < -1.5 \)

### Conclusion
From the comparisons:
- \( x = 1.4 \) is less than \( y = 1.8 \) but greater than \( y = -1.5 \).
- \( x = -2 \) is less than both values of \( y \).

Since we have contradictory results, we conclude that there is no consistent relationship between \( x \) and \( y \).

### Final Answer
The answer is that there is no relation between \( x \) and \( y \).

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