HesapHanem

System of Three Equations Solver

Enter the coefficients and constants of three equations to solve for x, y and z.

How it is calculated

The system is written as aix + biy + ciz = di for rows i = 1, 2, 3. The tool uses Cramer's rule: with D the determinant of the 3 × 3 coefficient matrix, x = Dx / D, y = Dy / D, z = Dz / D, where Dx, Dy, Dz are the same determinant with the x, y or z column replaced by the constants d. All twelve fields must be filled in; type 0 for a missing term, since a blank field returns no result. Results are rounded to four decimals, and D is shown rounded too, so a system with very small coefficients can display D as 0.0000 while still being solved. If D is zero relative to the size of the rows, the equations have no single solution and the tool shows nothing.

Worked example

Solve x + y + z = 6, 2x − y + z = 3 and x + 2y − z = 2. The coefficient determinant is D = 1(1 − 2) − 1(−2 − 1) + 1(4 + 1) = −1 + 3 + 5 = 7. Replacing the x column by 6, 3, 2 gives Dx = 7, the y column gives Dy = 14, and the z column gives Dz = 21. So x = 7/7 = 1.0000, y = 14/7 = 2.0000 and z = 21/7 = 3.0000, with D = 7.0000. Check in the first equation: 1 + 2 + 3 = 6.

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Common mistakes

  • Leaving a coefficient blank when a variable is absent from an equation. A blank gives no result; enter 0 for the missing term.
  • Typing the equation in a different order than the columns. Every row must follow the same x, y, z order, and the constant goes in the last box on the right side of the equals sign.
  • Forgetting to move all variables to the left side first. An equation like 2x = y + 5 must be entered as 2x − y + 0z = 5.
  • Dropping a sign: a coefficient of −z is −1, not 1. One wrong sign changes D and every answer.
  • Expecting an answer when one equation is a multiple of, or a sum of, the others. Then D is zero, and the tool gives no result because there is no unique solution: either no solution or infinitely many.

About this calculator

Cramer's rule solves a system of three linear equations in three unknowns by expressing each variable as a ratio of determinants: x = Dx/D, y = Dy/D, and z = Dz/D, where D is the determinant of the coefficient matrix and Dx, Dy, Dz replace the corresponding column with the constants column. If D = 0, the system has no unique solution — it's either inconsistent or has infinitely many solutions.

This solver takes the nine coefficients and three constants, computes all four determinants, and returns x, y, and z along with the coefficient determinant so you can confirm the system was actually solvable. Engineering, physics, and economics students use it to check circuit-analysis, statics, or input-output model homework, while professionals use the same method to solve small systems (three-force equilibrium, three-component mixture ratios) without setting up a full matrix solver.

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Comments (1)

  • Editorial team

    When the determinant is zero or extremely close to it, the tool shows no result, and it cannot tell you whether the system has no solution or infinitely many, so eliminate a variable by hand to find out. Otherwise, put the displayed x, y and z back into your equations. Because they are rounded to four decimals, the last digit may differ slightly. Which kind of problem are you solving with this?

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