Simultaneous Equations
Solve two equations instantly.
Core Concepts
When you have two equations like 2x + y = 5 and 3x + 2y = 8, you can solve them using the Cross-Multiplication method (Paravartya).
The Formula
For ax + by = m and cx + dy = n:
x = (bn - dm) / (bc - ad)
(Or simply: Cross-multiply Y coefficients with constants, subtract, and divide by determinant).
Addition/Subtraction Trick
If coefficients are symmetric (like 2x+3y=7 and 3x+2y=8):
1. Add equations to get x+y.
2. Subtract equations to get x-y.
3. Solve for x and y easily.
Solved Examples
Example 1: x + y = 5, x - y = 1
Question: Solve for x and y.
• Add equations: (x+x) + (y-y) = 5+1.
• 2x = 6 → x = 3.
• Put x=3 in first eq: 3 + y = 5 → y = 2.
Example 2: 2x + y = 7, x + y = 4
Question: Solve for x and y.
• Subtract eq2 from eq1.
• (2x-x) + (y-y) = 7-4.
• x = 3.
• 3 + y = 4 → y = 1.
• Answer: x=3, y=1.
Example 3: 2x + 3y = 13, 3x + 2y = 12
Question: Solve for x and y.
• Add: 5x + 5y = 25 → x + y = 5.
• Subtract: -x + y = 1 → y - x = 1.
• Add (x+y) and (y-x): 2y = 6 → y = 3.
• x + 3 = 5 → x = 2.
• Answer: x=2, y=3.
Example 4: 2x + y = 5, 3x + 2y = 8
Question: Solve for x and y.
• Multiply eq1 by 2: 4x + 2y = 10.
• Subtract eq2: (4x-3x) + (2y-2y) = 10-8.
• x = 2.
• 2(2) + y = 5 → 4 + y = 5 → y = 1.
• Answer: x=2, y=1.
