I. ` 2x+3y=14`
II. ` 4x+2y=16`

Options

A. if ` x ge y,`i.e., x is greater than or equal to y.
B. if `x gt y`, i.e., x is greater than y.
C. if ` x le y`, i.e., x is less than or equal to y.
D. if ` x lt y`, i.e., x is less than y.

Explanation

To solve the given system of equations:

I. \( 2x + 3y = 14 \)
II. \( 4x + 2y = 16 \)

we will use the elimination method.

### Step 1: Make the coefficients of \( x \) the same

To eliminate \( x \), we can multiply the first equation by 2 so that the coefficient of \( x \) in both equations becomes 4.

\[ 2(2x + 3y) = 2(14) \]

This simplifies to:

\[ 4x + 6y = 28 \quad \text{(Equation III)} \]

### Step 2: Write down the second equation

The second equation remains as it is:

\[ 4x + 2y = 16 \quad \text{(Equation II)} \]

### Step 3: Subtract the second equation from the first

Now, we will subtract Equation II from Equation III:

\[ (4x + 6y) - (4x + 2y) = 28 - 16 \]

This simplifies to:

\[ 4y = 12 \]

### Step 4: Solve for \( y \)

Now, divide both sides by 4:

\[ y = \frac{12}{4} = 3 \]

### Step 5: Substitute \( y \) back into one of the original equations to find \( x \)

We can substitute \( y = 3 \) into Equation II:

\[ 4x + 2(3) = 16 \]

This simplifies to:

\[ 4x + 6 = 16 \]

### Step 6: Solve for \( x \)

Now, subtract 6 from both sides:

\[ 4x = 16 - 6 \]

This gives us:

\[ 4x = 10 \]

Now, divide by 4:

\[ x = \frac{10}{4} = \frac{5}{2} = 2.5 \]

### Step 7: State the solution

Thus, the solution to the system of equations is:

\[ x = 2.5, \quad y = 3 \]

### Step 8: Determine the relationship between \( x \) and \( y \)

Since \( x = 2.5 \) and \( y = 3 \), we can conclude that \( x < y \).

### Final Answer

The correct relation is \( x < y \).

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