Simultaneous Equations Calculator (2 and 3 Variables) with Cramer’s Rule
Type each equation on its own line as its coefficients followed by the constant. You get x, y (and z) as exact fractions with the determinants used.
- Nothing you type is stored
- Free
- No signup
Solution
x = 1, y = 2
- x
- 1
- y
- 2
Solved by Cramer’s rule: each unknown = its determinant ÷ the main determinant D.
Show working
- 2x + 3y = 8
- x − y = −1
- D = a₁b₂ − a₂b₁ = 2·(−1) − 1·3 = −5
- Dx = −5 → x = Dx / D = −5 / (−5) = 1
- Dy = −10 → y = Dy / D = −10 / (−5) = 2
How to use
- Choose 2 equations (x, y) or 3 equations (x, y, z).
- Type one equation per line: for 2x + 3y = 8 type “2 3 8”.
- Read the solution and open “Show working” for D, Dx, Dy (and Dz).
Good to know
Cramer’s rule: x = Dx/D, y = Dy/D (and z = Dz/D), where D is the determinant of the coefficients and Dx is the same determinant with the x-column replaced by the constants.
If D = 0 the equations have no unique solution: the lines (or planes) are parallel or the same, so there are either no solutions or infinitely many. NCERT Class 10 calls these inconsistent or dependent pairs.
Write missing terms as 0: “y + 5z = −4” is “0 1 5 −4”. Negative numbers and decimals are fine; separate numbers with spaces or commas.
Questions
How do I enter 3x − y = 7?
Type “3 -1 7” on one line: the coefficient of x, the coefficient of y, then the number on the right.
What is Cramer’s rule?
A way to solve linear equations using determinants: each unknown equals a determinant with one column replaced by the constants, divided by the main determinant.
What if the calculator says no unique solution?
The main determinant is 0. The equations are either contradictory (parallel lines) or describe the same line; check the coefficients.
Is this the same as elimination or substitution?
It gives the same answer. Cramer’s rule is just a neat formula version, handy for 2×2 and 3×3 systems.
Is my data stored or sent anywhere?
No. The calculation runs entirely in your browser; nothing you type leaves your device.