Two equations (I) and (II) are given in each question. On the basis of these equations you have to decide the relation between x and y and give answer
I. `7 x - 3 y = 13`
II. ` 5 x + 4y = 40`

Options

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

Answer

To solve the given equations and determine the relationship between \( x \) and \( y \), we will follow these steps:

### Step 1: Write down the equations
We have two equations:
1. \( 7x - 3y = 13 \) (Equation I)
2. \( 5x + 4y = 40 \) (Equation II)

### Step 2: Eliminate one variable
To eliminate \( y \), we need to make the coefficients of \( y \) the same in both equations. The least common multiple (LCM) of 3 and 4 is 12.

We will multiply Equation I by 4 and Equation II by 3:

- Multiply Equation I by 4:
\[ 4(7x - 3y) = 4(13) \implies 28x - 12y = 52 \quad \text{(Equation III)} \]

- Multiply Equation II by 3:
\[ 3(5x + 4y) = 3(40) \implies 15x + 12y = 120 \quad \text{(Equation IV)} \]

### Step 3: Add the two new equations
Now we will add Equation III and Equation IV to eliminate \( y \):
\[ (28x - 12y) + (15x + 12y) = 52 + 120 \]
This simplifies to:
\[ 28x + 15x = 172 \]
\[ 43x = 172 \]

### Step 4: Solve for \( x \)
Now, divide both sides by 43:
\[ x = \frac{172}{43} = 4 \]

### Step 5: Substitute \( x \) back to find \( y \)
Now that we have \( x = 4 \), we can substitute this value back into either of the original equations to find \( y \). We will use Equation II:
\[ 5(4) + 4y = 40 \]
This simplifies to:
\[ 20 + 4y = 40 \]
Subtract 20 from both sides:
\[ 4y = 20 \]
Now, divide by 4:
\[ y = 5 \]

### Step 6: Determine the relationship between \( x \) and \( y \)
We have found \( x = 4 \) and \( y = 5 \). Thus, the relationship is:
\[ x < y \]

### Conclusion
The final answer is that \( x \) is less than \( y \).

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