Calculator for System of Equations

Formula

For a₁x + b₁y = c₁ and a₂x + b₂y = c₂, D = a₁b₂ − a₂b₁. If D ≠ 0, x = (c₁b₂ − c₂b₁) ÷ D and y = (a₁c₂ − a₂c₁) ÷ D. If D = 0, dependent constraints wey agree get infinitely many solutions; constraints wey contradict no get any. Equation wey coefficients na zero but constant no be zero na contradiction.

Example wey show the calculation

2x + y = 7 and x − y = 2: D = 2 × (−1) − 1 × 1 = −3. Then x = (7 × (−1) − 2 × 1) ÷ (−3) = (−9) ÷ (−3) = 3, and y = (2 × 2 − 1 × 7) ÷ (−3) = (−3) ÷ (−3) = 1. Check: 2 × 3 + 1 = 7, and 3 − 1 = 2. Both equations dey correct, so (3, 1) na di only point where di two lines cross.

Where e fit work and limits

Dis page dey solve two linear equations with two unknowns, including identity and contradiction rows wey coefficients na zero. E no dey solve nonlinear terms, inequalities or bigger systems. Results dey show up to 12 significant digits. Equations wey nearly depend on each other fit dey very sensitive to input precision. Values outside di supported numeric range go show error.

Version of di formula: e95780d58fb5 · Example don get check: 2026-10-31

Questions wey people dey ask often

Wetin the solution mean for geometry?

Linear equation wey no degenerate dey describe line. Two lines wey cross get one solution; separate parallel lines no get any; di same line twice get infinitely many. Equation wey coefficients na zero instead mean say no extra restriction dey or contradiction dey.

Wetin if determinant na zero?

No single unique pair dey. Di page dey check whether di constraints agree, including rows wey coefficients na zero. Dependent equations wey agree get infinitely many solutions; contradiction like zero equalling nonzero constant no get any.

Why dem dey call am Cramer's rule?

Na Gabriel Cramer dem name am after; e publish di determinant method for 1750. Na di same elimination wey you for do by hand: multiply, subtract, solve, but dem pack am inside two formulas.

I fit solve am by substitution instead?

Yes, and you suppose get di same answer. From x − y = 2, x = y + 2; put dat inside 2x + y = 7 and you go get 2(y + 2) + y = 7, so 3y = 3, y = 1, x = 3. Substitution dey easier by hand for simple systems; determinants dey scale better.

Both equations must be linear?

Yes, for dis method. If any equation get x², y², xy, or square root of x or y, di determinant formulas no apply. Di system fit get zero, one, two or more solutions.

How I fit check my answer?

Put x and y back inside both original equations, just as di check chips dey do. Correct solution go make both sides equal for both equations. Solution wey satisfy only one equation no correct.

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