In the following question two equations are given in variables x and y. you have to solve these equations and determine the relation between x and y.
I. `6x^(2) + 13x + 6 = 0`
`15y^(2) + 13y + 2 = 0`

Options

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

Answer

To solve the given equations and determine the relationship between \(x\) and \(y\), we will follow these steps:

### Step 1: Solve the first equation for \(x\)
The first equation is:
\[ 6x^2 + 13x + 6 = 0 \]
To factor this equation, we can rewrite it:
\[ 6x^2 + 9x + 4x + 6 = 0 \]
Next, we group the terms:
\[ (6x^2 + 9x) + (4x + 6) = 0 \]
Factoring out common terms:
\[ 3x(2x + 3) + 2(2x + 3) = 0 \]
Now, we can factor by grouping:
\[ (3x + 2)(2x + 3) = 0 \]
Setting each factor to zero gives us:
1. \(3x + 2 = 0 \Rightarrow x = -\frac{2}{3}\)
2. \(2x + 3 = 0 \Rightarrow x = -\frac{3}{2}\)

### Step 2: Solve the second equation for \(y\)
The second equation is:
\[ 15y^2 + 13y + 2 = 0 \]
We can factor this equation:
\[ 15y^2 + 10y + 3y + 2 = 0 \]
Grouping the terms:
\[ (15y^2 + 10y) + (3y + 2) = 0 \]
Factoring out common terms:
\[ 5y(3y + 2) + 1(3y + 2) = 0 \]
Factoring by grouping gives us:
\[ (5y + 1)(3y + 2) = 0 \]
Setting each factor to zero gives us:
1. \(5y + 1 = 0 \Rightarrow y = -\frac{1}{5}\)
2. \(3y + 2 = 0 \Rightarrow y = -\frac{2}{3}\)

### Step 3: Compare the values of \(x\) and \(y\)
Now we have the values:
- For \(x\): \(x_1 = -\frac{2}{3}\) and \(x_2 = -\frac{3}{2}\)
- For \(y\): \(y_1 = -\frac{1}{5}\) and \(y_2 = -\frac{2}{3}\)

Next, we compare these values:
- Comparing \(y_1 = -\frac{1}{5}\) with \(x_1 = -\frac{2}{3}\):
\(-\frac{1}{5} > -\frac{2}{3}\) (since \(-0.2 > -0.666\))

- Comparing \(y_1 = -\frac{1}{5}\) with \(x_2 = -\frac{3}{2}\):
\(-\frac{1}{5} > -\frac{3}{2}\) (since \(-0.2 > -1.5\))

- Comparing \(y_2 = -\frac{2}{3}\) with \(x_1 = -\frac{2}{3}\):
\(-\frac{2}{3} = -\frac{2}{3}\)

- Comparing \(y_2 = -\frac{2}{3}\) with \(x_2 = -\frac{3}{2}\):
\(-\frac{2}{3} > -\frac{3}{2}\)

### Conclusion
From the comparisons, we can conclude that:
- \(y\) is always greater than or equal to \(x\).

Thus, the relationship between \(x\) and \(y\) is:
\[ \text{y is greater than or equal to x} \]