System of Linear Equations Solver
Solve a system of two linear equations in two unknowns (x and y) using Cramer's Rule. Use it when you need exact solutions to simultaneous equations in algebra, physics, or engineering problems.
About this calculator
A 2×2 linear system has the form a₁x + b₁y = c₁ and a₂x + b₂y = c₂. The solution uses Cramer's Rule, which relies on determinants. The main determinant is D = a₁·b₂ − a₂·b₁. The solution for x is Dₓ = (c₁·b₂ − c₂·b₁)/D, and for y is Dᵧ = (a₁·c₂ − a₂·c₁)/D. If D = 0 and the numerators are also zero, the system has infinitely many solutions (dependent equations). If D = 0 but a numerator is non-zero, the system is inconsistent — the lines are parallel and there is no solution. When D ≠ 0, exactly one solution exists and the lines intersect at a single point. Cramer's Rule directly implements the elimination method in formula form, making it efficient and easy to verify by hand.
How to use
Solve the system: 2x + 3y = 12 and x − y = 1. Enter a₁ = 2, b₁ = 3, c₁ = 12, a₂ = 1, b₂ = −1, c₂ = 1. Main determinant: D = 2×(−1) − 1×3 = −2 − 3 = −5. For x: Dₓ = (12×(−1) − 1×3)/(−5) = (−12 − 3)/(−5) = −15/−5 = 3. For y: Dᵧ = (2×1 − 1×12)/(−5) = (2 − 12)/(−5) = −10/−5 = 2. Solution: x = 3, y = 2. Verify: 2(3) + 3(2) = 12 ✓ and 3 − 2 = 1 ✓.
Frequently asked questions
What does it mean when the determinant of a system of equations is zero?
The determinant D = a₁·b₂ − a₂·b₁ equals zero when the two equations represent lines that are either parallel or identical. If D = 0 and the numerator determinants Dₓ and Dᵧ are also zero, the equations are dependent — they describe the same line and there are infinitely many solutions. If D = 0 but at least one numerator is non-zero, the equations are inconsistent — the lines are parallel but distinct, and there is no point satisfying both simultaneously. In either case Cramer's Rule breaks down because division by zero is undefined, so you must recognize these cases before applying the formula.
How is Cramer's Rule different from the substitution and elimination methods?
Substitution involves isolating one variable in one equation and replacing it in the other, which is intuitive but can involve messy fractions. Elimination adds or subtracts multiples of the equations to cancel one variable directly. Cramer's Rule computes both unknowns at once using determinant formulas without rearranging equations step by step, making it especially systematic and easy to program or verify. For 2×2 systems all three methods give the same answer; Cramer's Rule becomes particularly elegant when implemented in a calculator because it is a single formula per variable. For larger systems (3×3 and beyond), Gaussian elimination is generally preferred for efficiency.
How do you check if your solution to a system of linear equations is correct?
Substitute the values of x and y you found back into both original equations and verify that each equation becomes a true statement. For example, if you found x = 3 and y = 2 for the system 2x + 3y = 12 and x − y = 1, check: 2(3) + 3(2) = 6 + 6 = 12 ✓ and 3 − 2 = 1 ✓. If either equation fails, recheck your determinant calculations for sign errors — a frequent source of mistakes. This verification step takes only seconds and guarantees the answer is correct before you use it in a larger problem.