Math
System of Equations Calculator
Solve 2x2 or 3x3 linear systems with Cramer's rule. Get unique solutions for x, y, and z, or a clear answer when the system has no solution or infinite solutions.
System of equations
x +y =
x +y =
Unique solution
x
3
y
2
Determinant: -5
Frequently Asked Questions about the System of Equations Calculator
How does Cramer's rule solve a system of equations?
Cramer's rule expresses each variable as a ratio of two determinants. For a 2x2 system a1*x + b1*y = c1 and a2*x + b2*y = c2, the main determinant is D = a1*b2 - a2*b1. Then x = (c1*b2 - c2*b1) / D and y = (a1*c2 - a2*c1) / D. For a 3x3 system the same logic applies: replace the column of the variable you want with the constants column and divide by the 3x3 main determinant. This calculator applies those formulas directly and reports the solution along with the value of the main determinant.
What does it mean when the determinant is 0?
A determinant of 0 means the coefficient matrix is singular and no single unique solution exists. The calculator then checks the augmented determinants to distinguish two cases. If every substituted determinant is also 0, the equations are dependent and the system has infinite solutions (the lines or planes coincide). If any substituted determinant is non-zero, the system is inconsistent and has no solution (the lines are parallel, or the planes never share a common point).
When should I use Cramer's rule, substitution, or elimination?
Cramer's rule is a good fit for 2x2 and 3x3 systems when you want a direct closed-form formula for each variable. Substitution is quick when one equation is already isolated for a variable or has a coefficient of 1 (for example, y = 3x + 2). Elimination scales better beyond 3x3 and makes it easy to spot dependent or inconsistent systems early. All three methods give the same unique solution when one exists.
Where do systems of equations show up in real life?
Anywhere two or more quantities are tied together by linear constraints. Typical examples include splitting a $1,500 budget across two expense categories, mixing a 10% and a 30% saline solution to hit a 20% target, applying Kirchhoff's laws to find current in a circuit, setting ticket prices when you know the number sold and total revenue, and finding the intersection of supply and demand curves in economics.
Can this calculator handle decimals and negative numbers?
Yes. You can enter any real number as a coefficient or constant, including decimals and negative values. Fractions like 1/3 should be entered as their decimal equivalent (0.333333), since the inputs are read as plain decimals. The calculator carries full floating-point precision throughout and rounds each final result to six decimal places to suppress floating-point noise.