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.
A) 5x + 2y = 31
B) 3x + 7y = 36

Options

A. `X gt Y`
B. `X ge Y`
C. `X lt Y`
D. `X le Y`

Explanation

To solve the equations and determine the relationship between \(X\) and \(Y\), we will follow these steps:

### Step 1: Write down the equations
We have the following two equations:
1. \(5x + 2y = 31\) (Equation 1)
2. \(3x + 7y = 36\) (Equation 2)

### Step 2: Multiply the equations to eliminate one variable
To eliminate \(y\), we can multiply Equation 1 by \(7\) and Equation 2 by \(2\) to make the coefficients of \(y\) the same.

- Multiply Equation 1 by \(7\):
\[ 7(5x + 2y) = 7(31) \implies 35x + 14y = 217 \quad \text{(Equation 3)} \]

- Multiply Equation 2 by \(2\):
\[ 2(3x + 7y) = 2(36) \implies 6x + 14y = 72 \quad \text{(Equation 4)} \]

### Step 3: Subtract the equations to eliminate \(y\)
Now, we can subtract Equation 4 from Equation 3:
\[ (35x + 14y) - (6x + 14y) = 217 - 72 \]
This simplifies to:
\[ 29x = 145 \]

### Step 4: Solve for \(x\)
Now, divide both sides by \(29\):
\[ x = \frac{145}{29} = 5 \]

### Step 5: Substitute \(x\) back to find \(y\)
Now that we have \(x\), we can substitute it back into either of the original equations to find \(y\). Let's use Equation 1:
\[ 5(5) + 2y = 31 \]
This simplifies to:
\[ 25 + 2y = 31 \]
Subtract \(25\) from both sides:
\[ 2y = 6 \]
Now divide by \(2\):
\[ y = 3 \]

### Step 6: Determine the relationship between \(x\) and \(y\)
Now we have \(x = 5\) and \(y = 3\). Therefore, we can conclude that:
\[ x > y \]

### Final Answer
The relationship between \(X\) and \(Y\) is \(X > Y\).

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