` 5x + 2y = 31`
` 3x + 7y = 36`

Options

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

Explanation

To solve the system of equations given by:

1. \( 5x + 2y = 31 \) (Equation 1)
2. \( 3x + 7y = 36 \) (Equation 2)

we will use the method of elimination.

### Step 1: Multiply the equations to align coefficients
To eliminate one of the variables, we can multiply the equations by suitable numbers. We will multiply Equation 1 by 3 and Equation 2 by 5:

- Multiply Equation 1 by 3:
\[ 3(5x + 2y) = 3(31) \]
\[ 15x + 6y = 93 \] (Equation 3)

- Multiply Equation 2 by 5:
\[ 5(3x + 7y) = 5(36) \]
\[ 15x + 35y = 180 \] (Equation 4)

### Step 2: Subtract the equations
Now, we will subtract Equation 3 from Equation 4 to eliminate \(x\):

\[ (15x + 35y) - (15x + 6y) = 180 - 93 \]
This simplifies to:
\[ 35y - 6y = 180 - 93 \]
\[ 29y = 87 \]

### Step 3: Solve for \(y\)
Now, we can solve for \(y\):
\[ y = \frac{87}{29} \]
\[ y = 3 \]

### Step 4: Substitute \(y\) back to find \(x\)
Now that we have \(y\), we can substitute it back into Equation 1 to find \(x\):
\[ 5x + 2(3) = 31 \]
\[ 5x + 6 = 31 \]
\[ 5x = 31 - 6 \]
\[ 5x = 25 \]
\[ x = \frac{25}{5} \]
\[ x = 5 \]

### Step 5: State the values of \(x\) and \(y\)
We have found:
- \( x = 5 \)
- \( y = 3 \)

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

### Final Answer
Thus, the solution to the equations is:
- \( x = 5 \)
- \( y = 3 \)
- The relation is \( x > y \).

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