In each of these questions, two equations (I) and (II) are given . You have to solve both the equations and give answer
I. ` 2 x + 5y = 6 `
II.` 5 x + 11 y = 9 `

Options

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

Answer

To solve the given equations step by step, we will follow the method of elimination. The equations are:

I. \( 2x + 5y = 6 \)
II. \( 5x + 11y = 9 \)

### Step 1: Multiply the equations to equalize the coefficients of \( y \)

We will multiply the first equation by \( 11 \) and the second equation by \( 5 \) to make the coefficients of \( y \) the same.

- Multiply Equation I by \( 11 \):
\[ 11(2x + 5y) = 11(6) \implies 22x + 55y = 66 \quad \text{(Equation III)} \]

- Multiply Equation II by \( 5 \):
\[ 5(5x + 11y) = 5(9) \implies 25x + 55y = 45 \quad \text{(Equation IV)} \]

### Step 2: Subtract the two new equations

Now we will subtract Equation IV from Equation III to eliminate \( y \):

\[ (22x + 55y) - (25x + 55y) = 66 - 45 \]

This simplifies to:

\[ 22x - 25x = 66 - 45 \]

\[ -3x = 21 \]

### Step 3: Solve for \( x \)

Now, we solve for \( x \):

\[ x = \frac{21}{-3} = -7 \]

### Step 4: Substitute \( x \) back into one of the original equations

We will substitute \( x = -7 \) back into Equation I to find \( y \):

\[ 2(-7) + 5y = 6 \]

This simplifies to:

\[ -14 + 5y = 6 \]

### Step 5: Solve for \( y \)

Now, isolate \( y \):

\[ 5y = 6 + 14 \]
\[ 5y = 20 \]
\[ y = \frac{20}{5} = 4 \]

### Final Solution

The solution to the equations is:
\[ x = -7, \quad y = 4 \]

### Step 6: Compare the values of \( x \) and \( y \)

Now, we can compare \( x \) and \( y \):
- Since \( -7 < 4 \), we conclude that \( x < y \).