2x2 System of Equations Solver
Solve a two-equation linear system and verify both resulting values.
Find the intersection of two linear equations
A two-by-two linear system contains two equations and two unknown values. When the coefficient determinant is nonzero, the lines have one intersection and Cramer's rule provides that point directly. Enter each equation in coefficient form, and the solver reports x, y, the determinant, and substitutions back into both original left sides. Those checks make a transcription error easier to spot than an isolated answer would.
The defaults represent x plus y equals 5 and x minus y equals 1. Their determinant is negative 2, and the unique solution is x equals 3 and y equals 2. Substitution returns 5 for the first equation and 1 for the second. If the determinant becomes zero, the page reports that there is no unique solution rather than dividing by zero.
Private arithmetic with visible checks
Every coefficient and calculation stays in your browser. This is an exact algebraic setup evaluated with JavaScript floating-point numbers, so heavily scaled or nearly dependent equations can show rounding effects. Keep units consistent, preserve signs when entering coefficients, and use the displayed substitution lines to confirm that the solution satisfies the values you intended to enter.
Frequently Asked Questions
What equation form does this solver use?
Enter coefficients for a1x plus b1y equals c1 and a2x plus b2y equals c2.
What does a zero determinant mean?
A zero determinant means the two equations do not have one unique intersection point.
How are the values calculated?
The solver applies Cramer's rule to the coefficient determinant and the two replacement determinants.
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