Simultaneous Equations Solver (2x2)
This solver finds the solution to a system of two linear equations with two unknowns. It uses Cramer's rule, a determinant-based method to calculate the values of x and y that satisfy both equations simultaneously. When two lines intersect, their point of intersection (x, y) satisfies both equations. This solver displays the step-by-step working using determinants, making it educational and useful for verification. Simultaneous equations appear frequently in mathematics, science, engineering, and economics whenever you need to solve multiple constraints at once.",
Cramer's rule for 2x2 systems
System:
a1x + b1y = c1
a2x + b2y = c2
Determinants:
D = a1*b2 - a2*b1
Dx = c1*b2 - c2*b1
Dy = a1*c2 - a2*c1
Solution (if D != 0):
x = Dx / D
y = Dy / D
Simultaneous equations: frequently asked questions
What are simultaneous equations?
Simultaneous equations (or systems of equations) are sets of equations that must all be satisfied at the same time. For a 2x2 system, you have two equations with two unknowns, typically x and y. The solution is the point (x, y) that satisfies both equations.
What is Cramer's rule?
Cramer's rule is a mathematical formula for solving a system of linear equations using determinants. For the system a1x + b1y = c1 and a2x + b2y = c2, the determinant D = a1*b2 - a2*b1. Then x = (c1*b2 - c2*b1)/D and y = (a1*c2 - a2*c1)/D.
What does it mean if the determinant is zero?
If D = 0, the system has no unique solution. This occurs when the two equations represent parallel lines (no solution) or the same line (infinitely many solutions). The solver will display an error if D = 0.
How do I use this solver?
Enter the coefficients a1, b1, c1 for the first equation (a1x + b1y = c1) and a2, b2, c2 for the second equation (a2x + b2y = c2). The solver calculates x and y using Cramer's rule and shows the step-by-step working.
What is the difference between 'no solution' and 'infinitely many solutions'?
If the equations represent parallel lines, there is no solution (they never intersect). If the equations represent the same line, there are infinitely many solutions (every point on the line satisfies both equations). Both cases occur when D = 0.
Methodology
Reviewed by the CalculatorHub team, edited by James Graham, 14 June 2026. See our methodology.