In the following questions two equation numbered I and II are given You have to solve both equations and give answer
(ii) `5x-2y=31`
(ii) `3x+7y=36`

Options

A. if x > y
B. if x ≥ y
C. if x < y
D. if x ≤ y

Answer

To solve the given equations step by step, let's start with the equations provided:

1. **First Equation:** \( 5x - 2y = 31 \) (Equation I)
2. **Second Equation:** \( 3x + 7y = 36 \) (Equation II)

### Step 1: Multiply the equations to align coefficients
To eliminate one variable, we can multiply the equations by suitable numbers so that the coefficients of \( x \) or \( y \) become equal. Here, we will multiply Equation I by 3 and Equation II by 5.

- Multiply Equation I by 3:
\[ 3(5x - 2y) = 3(31) \implies 15x - 6y = 93 \quad \text{(Equation III)} \]

- Multiply Equation II by 5:
\[ 5(3x + 7y) = 5(36) \implies 15x + 35y = 180 \quad \text{(Equation IV)} \]

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

\[ (15x + 35y) - (15x - 6y) = 180 - 93 \]
This simplifies to:
\[ 35y + 6y = 87 \implies 41y = 87 \]

### Step 3: Solve for \( y \)
Now, we can solve for \( y \):
\[ y = \frac{87}{41} \]

### Step 4: Substitute \( y \) back to find \( x \)
Next, we substitute \( y \) back into one of the original equations to find \( x \). We'll use Equation I:
\[ 5x - 2\left(\frac{87}{41}\right) = 31 \]
This simplifies to:
\[ 5x - \frac{174}{41} = 31 \]
To eliminate the fraction, we can multiply the entire equation by 41:
\[ 41(5x) - 174 = 31 \times 41 \]
This gives:
\[ 205x - 174 = 1271 \]
Now, add 174 to both sides:
\[ 205x = 1271 + 174 \implies 205x = 1445 \]
Now, divide by 205:
\[ x = \frac{1445}{205} = \frac{289}{41} \]

### Step 5: Compare \( x \) and \( y \)
Now we have the values:
- \( x = \frac{289}{41} \)
- \( y = \frac{87}{41} \)

To compare \( x \) and \( y \):
\[ \frac{289}{41} > \frac{87}{41} \]

Thus, \( x > y \).

### Final Answer
The answer is that \( x \) is greater than \( y \), which corresponds to **Option 1**.

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