Skip to content
Quick Tools
Menu
School

Systems of Equations Calculator (2x2 and 3x3)

This systems of equations calculator solves 2x2 and 3x3 linear systems, shows elimination and Cramer's rule steps, and flags no solution or infinitely many solutions.

System size

Integers, decimals, and fractions such as 3/4. Each row is one linear equation.

Solution

One solution: x = 1, y = 2.

  • x = 1
  • y = 2

Elimination

  1. Write the system.
  2. (1) 2x + 3y = 8
  3. (2) x − y = −1
  4. Eliminate x in equation (2). Multiply equation (1) by 1/2 and subtract it from equation (2): x − y = −1 minus (x + (3/2)y = 4) gives −(5/2)y = −5.
  5. Solve (2) for y: −(5/2)y = (−5), so y = (−5) ÷ (−5/2) = 2.
  6. Substitute 3y = 6 into (1): 2x = 2, so x = 2 ÷ 2 = 1.
  7. Solution: x = 1, y = 2.

Cramer's rule

  1. Coefficient determinant D = (2)(−1) − (1)(3) = −5.
  2. Dx replaces the x column with the constants: Dx = (8)(−1) − (−1)(3) = −5.
  3. Dy replaces the y column with the constants: Dy = (2)(−1) − (1)(8) = −10.
  4. x = Dx ÷ D = (−5) ÷ (−5) = 1.
  5. y = Dy ÷ D = (−10) ÷ (−5) = 2.
Graph of the two lines
  • 2x + 3y = 8
  • x − y = −1

The math stays in your browser.

How to use

  1. Choose 2×2 or 3×3.
  2. Enter each coefficient. Integers, decimals, and fractions such as 3/4 all work.
  3. Read the solution, or the message that the system has no solution or infinitely many solutions.
  4. Follow the elimination steps and the Cramer's rule determinants. A 2×2 system also draws the two lines.

Formula

For a1 x + b1 y = d1 and a2 x + b2 y = d2, D = a1 b2 − a2 b1. Cramer's rule is x = Dx/D and y = Dy/D, where Dx and Dy swap one column for the constants. Elimination multiplies equations so one variable cancels, then substitutes back. If D = 0, a false row such as 0 = 5 means no solution, and 0 = 0 means infinitely many. A 3×3 system uses the same tests with z.

Example. 2x + 3y = 8 and x − y = −1. Elimination gives y = 2 and x = 1. The determinants are D = −5, Dx = −5, and Dy = −10.

Example. x + y + z = 6, 2x − y + z = 3, and x + 2y − z = 2. Elimination and Cramer's rule both give x = 1, y = 2, and z = 3.

Frequently asked questions

What methods can solve a system?

Graphing, substitution, elimination, and matrix methods such as Cramer's rule or Gaussian elimination.

How do I know if there is no solution?

If elimination gives a false statement like 0 = 5, the lines are parallel and the system is inconsistent.

What about infinitely many solutions?

If elimination gives 0 = 0, the equations describe the same line or plane, so every point on it is a solution.

What is Cramer's rule?

Each variable equals a ratio of determinants. It only works when the coefficient determinant is not zero.

Can it solve nonlinear systems?

No. This tool handles linear equations only.

Related tools