Question

Consider the pair of equations: \(x^2 - xy - x = 22\) and \(y^2 - xy + y = 34\). If x > y, then x- y equals

• A. 8
• B. 6
• C. 7
• D. 4

Answer 1

The Correct Option is A

The system of equations is provided:

\[ x^2 - xy - x = 22 \quad \text{and} \quad y^2 - xy + y = 34 \]

Step 1: Sum the equations

Adding the two equations yields: \[ x^2 - xy - x + y^2 - xy + y = 56 \] This simplifies to: \[ x^2 - 2xy + y^2 - x + y = 56 \]

Step 2: Identify the quadratic structure

The left side of the equation can be expressed as: \[ (x - y)^2 - (x - y) = 56 \]

Step 3: Perform substitution

Let \( t = (x - y) \). The equation transforms to: \[ t^2 - t = 56 \] Rearranging gives: \[ t^2 - t - 56 = 0 \]

Step 4: Solve for \( t \)

Solving the quadratic equation: \[ (t - 8)(t + 7) = 0 \] The possible values for \( t \) are: \[ t = 8 \quad \text{or} \quad t = -7 \]

Step 5: Final deduction

Given that \( t = (x - y) \), the results are: \[ x - y = 8 \]