In the following questions, two equations numbered I and II are given. You have to solve both the equations and
Give answer If
`x^2-3x-40=0`
`y^2-15y+54=0`

Options

A. `xgty`
B. `xley`
C. `xgey`
D. x=y or the relationship cannot be established

Answer

To solve the given equations step by step, we will first solve each equation separately.

### Step 1: Solve the first equation \( x^2 - 3x - 40 = 0 \)

1. **Identify the equation**: We have \( x^2 - 3x - 40 = 0 \).
2. **Factor the quadratic equation**: We need to find two numbers that multiply to \(-40\) (the constant term) and add up to \(-3\) (the coefficient of \(x\)).
- The numbers are \(-8\) and \(5\) because \(-8 \times 5 = -40\) and \(-8 + 5 = -3\).
3. **Rewrite the equation**: We can factor the equation as:
\[ (x - 8)(x + 5) = 0 \]
4. **Set each factor to zero**:
- \(x - 8 = 0 \Rightarrow x = 8\)
- \(x + 5 = 0 \Rightarrow x = -5\)

So, the solutions for \(x\) are \(x = 8\) and \(x = -5\).

### Step 2: Solve the second equation \( y^2 - 15y + 54 = 0 \)

1. **Identify the equation**: We have \( y^2 - 15y + 54 = 0 \).
2. **Factor the quadratic equation**: We need to find two numbers that multiply to \(54\) (the constant term) and add up to \(-15\) (the coefficient of \(y\)).
- The numbers are \(-9\) and \(-6\) because \(-9 \times -6 = 54\) and \(-9 + -6 = -15\).
3. **Rewrite the equation**: We can factor the equation as:
\[ (y - 9)(y - 6) = 0 \]
4. **Set each factor to zero**:
- \(y - 9 = 0 \Rightarrow y = 9\)
- \(y - 6 = 0 \Rightarrow y = 6\)

So, the solutions for \(y\) are \(y = 9\) and \(y = 6\).

### Step 3: Compare the values of \(x\) and \(y\)

We have the following pairs of solutions:
1. \(x = -5\) and \(y = 6\) → Here, \(x < y\)
2. \(x = -5\) and \(y = 9\) → Here, \(x < y\)
3. \(x = 8\) and \(y = 6\) → Here, \(x > y\)
4. \(x = 8\) and \(y = 9\) → Here, \(x < y\)

From the comparisons, we can conclude:
- In two cases, \(x\) is less than \(y\).
- In one case, \(x\) is greater than \(y\).

Since there is no consistent relationship established between \(x\) and \(y\) across all cases, we conclude that the relationship cannot be established definitively.

### Final Answer:
The correct answer is option 4: The relationship cannot be established.

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