Two Linear Equations CalculatorFree online math calculator — instant results, no sign-up

For a₁x + b₁y = c₁ and a₂x + b₂y = c₂, define D = a₁b₂ − a₂b₁. If D ≠ 0, x = (c₁b₂ − c₂b₁)/D and y = (a₁c₂ − a₂c₁)/D. With the defaults: D = 2(−1) − 1(1) = −3; x = [5(−1) − 1(1)]/(−3) = 2; y = [2(1) − 1(5)]/(−3) = 1.

• xThe first unknown, whose default-example solution is 2.
• yThe second unknown, whose default-example solution is 1.
• a₁The first x coefficient, default 2.
• b₁The first y coefficient, default 1.
• c₁The first constant, default 5.
• a₂The second x coefficient, default 1.
• b₂The second y coefficient, default −1.
• c₂The second constant, default 1.
• DThe determinant of the coefficient matrix; a nonzero value means the system has exactly one solution.

Use the determinant method, also called Cramer's rule: calculate D from the four coefficients, then divide each unknown's numerator by D. For the default equations, D = −3, so a unique solution exists and the formulas give x = 2 and y = 1. Verify by substitution: 2(2) + 1 = 5 and 2 − 1 = 1.

These are the calculator's starting values and the results it actually produces for them.

1. 1Arrange both equations with the x and y terms on the left and the constant on the right, keeping signs when moving terms.
2. 2Enter the first equation's values: First x coefficient = 2, First y coefficient = 1, and First constant = 5.
3. 3Enter the second equation's values: Second x coefficient = 1, Second y coefficient = −1, and Second constant = 1.
4. 4Run the calculation and read the solution rows: x = 2 and y = 1.
5. 5Substitute both values into both original equations to confirm that each equality holds.

Further reading

• Khan Academy