Solving algebraic equations by cross multiplication | ex-2

Solving a Pair of Linear Equations Using Cross Multiplication

Given:

1) ax + by = a − b  
2) bx − ay = a + b
  

Step 1: Convert to Standard Form

Move all terms to one side to express both equations in standard form:

   First equation in standard form: ax + by - a + b = 0
   Second equation in standard form: bx − ay − a − b = 0

  

From this, we identify:

• a₁ = a, b₁ = b, c₁ = b − a
• a₂ = b, b₂ = −a, c₂ = −a − b

Step 2: Apply Cross Multiplication

By applying the cross multiplication method:

x / (b₁·c₂ − b₂·c₁) = y / (c₁·a₂ − c₂·a₁) = 1 / (a₁·b₂ − a₂·b₁)
  

Step 3: Substitute the Coefficients

• x numerator: b(−a − b) − (−a)(b − a) = −ab − b² + ab − a² = −b² − a²
• Numerator for y: Multiply (b − a) by b, then subtract the product of (−a − b) and a: (b − a) × b − (−a − b) × a = b² − ab + a² + ab = a² + b²
• Denominator: Multiply a with −a and subtract the product of b and b: a × (−a) − b² = −a² − b²

Step 4: Final Results

Now substitute into the formula:

x = (−b² − a²) / (−a² − b²) = 1  
y = (a² + b²) / (−a² − b²) = −1
  

Final Answer:

x = 1, y = −1