About 3 Variable System of Equations Solver
This solver finds x, y and z from three linear equations. The grid has three rows of four numbers, the three coefficients and the constant. The default example, x + y + z = 6, 2y + 5z = −4 and 2x + 5y − z = 27, gives x = 5, y = 3 and z = −2. It is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded.
It uses Gauss-Jordan elimination on the 3 × 4 augmented matrix. The steps list each row swap, scaling and subtraction, so you can follow them by hand. A zero in the coefficient position means the variable is missing from that equation: leave the cell blank.
Three planes can meet in a single point, along a line, or not at all. One point gives a unique answer as above. If a row reduces to 0 = c with c not zero, there is no solution. If a variable has no pivot, there are infinitely many solutions, and the solver reports that the equations are dependent.
How to use 3 Variable System of Equations Solver
- Pick a type: Linear, Quadratic, Cubic, Polynomial roots, 2 equations, 3 equations or Discriminant.
- For a single equation, type it in the box using ^ for powers, such as x^3 - 6x^2 + 11x - 6 = 0. The type is detected from what you enter.
- For a system, fill the grid with the coefficients and the constant of each equation, in the order x, y, z, constant.
- Read the roots or the solution at the top of the result.
- Open the steps to follow the working, and look at the graph to see where the curve meets the axis.
Features
- Linear, quadratic, cubic and higher-degree polynomial equations in one variable, with the degree detected automatically.
- Quadratic roots with the discriminant shown and explained, including complex roots in the form a + bi.
- Exact rational roots shown as integers or fractions, such as x = 1/2, and exact radical and complex roots such as x = 1 ± √2 or x = −1 ± 2i.
- Cubic and higher equations solved by the rational root theorem, reducing the degree step by step, with a numeric fallback when no rational root exists.
- Cubic discriminant reported, so you know whether to expect three real roots or one real and two complex.
- Systems of 2 and 3 linear equations solved by Gauss-Jordan elimination, with the augmented matrix and each row operation typeset.
- Detects systems with no solution, and gives a parametric solution such as x = 2 − t, y = t for systems with infinitely many.
- Brackets, fractions and rational equations through the expression engine; values that make a denominator zero are excluded.
- A graph of the left side minus the right side with the real roots marked, which you can zoom, pan and export.
- A standalone discriminant mode for b² − 4ac.
Reading the reduced matrix
When elimination finishes with the identity on the left, the last column is the solution.
[ 1 0 0 | x ]
[ 0 1 0 | y ]
[ 0 0 1 | z ]Worked examples
x + y + z = 6, 2y + 5z = −4, 2x + 5y − z = 27 gives x = 5, y = 3, z = −2
Check: 5 + 3 − 2 = 6, then 2(3) + 5(−2) = −4, and 2(5) + 5(3) − (−2) = 10 + 15 + 2 = 27.
Privacy: how it works
- Equations are solved by code running in your browser tab. They are not sent to Ulkaa or anyone else, and there is nothing to sign in to.
Tips
- Brackets, fractions and rational equations are accepted, so (x + 1)(x + 2) = 0, x/2 + 1 = 4 and (x + 1)/(x - 2) = 3 can be typed as they are.
- Move nothing yourself: write the equation as you have it, with terms on both sides if needed, and the solver collects them.
- Use a decimal or a fraction for coefficients, such as 0.5x + 2 = 5 or 1/2x + 2 = 5.
- In the system grid, leave a cell blank for a zero coefficient.
- For one variable, the equation can use x or any single letter, but only one letter per equation.
Frequently asked questions
Is it really free?
Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.
Do I need an account?
No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.
Does it run in my browser?
Yes. Your equations are processed by code running in your browser tab, on your own device.
Is my data uploaded?
No. Your equations are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.
How do I solve three equations with three unknowns?
Write the augmented matrix and row-reduce it to the identity on the left. The last column then holds x, y and z.
How do I enter an equation with a missing variable?
Leave that cell blank or type 0. For 2y + 5z = −4 the x coefficient is 0.
More Equation Solver pages
- Equation SolverSolve linear, quadratic, cubic and polynomial equations and systems of equations, with the working and a graph.
- Quadratic Equation SolverType a quadratic and get its roots, the discriminant and the working.
- Quadratic Formula CalculatorSee the quadratic formula applied to your numbers, line by line.
- Linear Equation SolverSolve for x in an equation with x on one or both sides.
- System of Equations CalculatorEnter the coefficients of two equations and get x and y with each row operation.
- Simultaneous Equations SolverSolve two equations at once and check the answer against both.
- Cubic Equation SolverFind all three roots of a cubic, real and complex.
- Polynomial Roots CalculatorFind every root of a polynomial of any degree and plot the real ones.
- Discriminant CalculatorEnter a, b and c to get b² − 4ac and the type of roots.