System of Equations Solver
Enter the coefficients for two linear equations (ax + by = e and cx + dy = f) to solve for x and y.
Last updated: September 5, 2026
Frequently asked questions
How do I solve a system of two linear equations?
Using Cramer's rule (what this calculator does): for ax+by=e and cx+dy=f, x = (ed−bf)/(ad−bc) and y = (af−ec)/(ad−bc), where ad−bc is the determinant of the coefficient matrix. This works as long as that determinant isn't zero.
What does it mean if there's no solution?
The two equations represent parallel lines (the same slope, different intercepts) that never cross - so there's no (x, y) pair that makes both equations true at the same time.
What does it mean if there are infinitely many solutions?
The two equations actually describe the exact same line (one might just be a multiple of the other) - every point on that line satisfies both equations, so there isn't one unique (x, y) answer.
Is a 'systems of equations calculator' the same as this solver?
Yes - "systems of equations solver" and "systems of equations calculator" both describe finding the (x, y) that satisfies two linear equations simultaneously, which is exactly what this tool does.
What does the determinant (ad − bc) actually represent geometrically?
It reflects whether the two lines have different slopes - a nonzero determinant means the lines intersect at exactly one point (a unique solution), while a zero determinant means the lines are parallel (either no solution or infinitely many, depending on the intercepts).
Can this solve equations with fractions or decimals as coefficients?
Yes - Cramer's rule works with any real number coefficients, including decimals and negative values, so equations like 2.5x + 1.2y = 7.4 work exactly like whole-number ones.
What's the solution to 2x + 3y = 16 and x − y = 2?
x = 4.4, y = 2.4 - using Cramer's rule: determinant = (2)(-1) − (3)(1) = -5, x = ((16)(-1) − (3)(2))/(-5) = (-22)/(-5) = 4.4, y = ((2)(2) − (16)(1))/(-5) = (-12)/(-5) = 2.4.