Simultaneous Equation Solver — 2 Equations, 2 Unknowns
Equation 1: a1*x + b1*y = c1
Equation 2: a2*x + b2*y = c2
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y =
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How to Solve Simultaneous Equations
Simultaneous equations are two or more equations that share the same unknowns. For a system of two linear equations with two unknowns (x and y), this calculator uses Cramer's rule, which relies on determinants to find the solution directly.
The determinant D = a1*b2 - a2*b1. If D is not zero, the system has a unique solution: x = (c1*b2 - c2*b1)/D and y = (a1*c2 - a2*c1)/D. If D = 0, the lines are either parallel (no solution) or identical (infinite solutions).
Simultaneous equations appear in physics (force equilibrium), economics (supply and demand), chemistry (mixture problems), and engineering (circuit analysis). They are a fundamental tool in applied mathematics.
Frequently Asked Questions
What is Cramer's rule?
Cramer's rule uses determinants to solve systems of linear equations. For two equations, it computes D = a1*b2 - a2*b1, then x = (c1*b2 - c2*b1)/D and y = (a1*c2 - a2*c1)/D. It works when D is not zero.
When does a system have no solution?
A system has no solution when the two equations represent parallel lines that never intersect. This occurs when the determinant is zero and the equations are not multiples of each other.
What if the determinant is zero?
If the determinant is zero, the lines are either parallel (no solution) or coincident (infinitely many solutions). The calculator checks which case applies.