Simultaneous Equations Calculator
Solve systems of linear equations with 2 or 3 variables step-by-step using Cramer’s Rule, elimination, or substitution methods instantly.
System of Equations Solver
Enter coefficients to solve for variables
Select how many variables your system contains
Solution & Steps
Variables, determinants, and step-by-step breakdown
Enter your coefficients and click Solve System to find the values of x, y, and z using Cramer’s Rule.
Solution Methods
Common mathematical methods used to solve systems of linear simultaneous equations.
| Method | Best Used For | Description |
|---|---|---|
| Substitution | Simple 2×2 systems | Isolate one variable in one equation and substitute it into the other. |
| Elimination | Most 2×2 systems | Add or subtract equations to eliminate one variable, then solve for the other. |
| Cramer’s Rule | Small systems (2×2, 3×3) | Uses matrix determinants to find exact variable values directly. |
| Gaussian Elimination | Larger systems (3×3+) | Converts the system to row-echelon form using matrix row operations. |
| Graphical Method | Visualizing 2×2 systems | Plots both equations on a graph; the intersection point is the solution. |
Simultaneous Equations FAQ
Everything you need to know about solving systems of linear equations.
Simultaneous equations are a set of equations containing multiple variables, where the solutions are the values that satisfy all the equations in the set at the same time. For example, if you have two variables (x and y), you need at least two independent equations to find a unique solution for both.
You can solve a 2×2 system using methods like Substitution, Elimination, or Cramer’s Rule. In the Elimination method, you multiply one or both equations by constants so that adding or subtracting them eliminates one variable, allowing you to solve for the other. This calculator uses Cramer’s Rule (determinants) to find the exact values of x and y instantly.
Cramer’s Rule is a mathematical technique for solving systems of linear equations using determinants. For a system ax + by = c and dx + ey = f, you calculate the main determinant D = ae – bd. Then, you find Dx and Dy by replacing the respective columns with the constants. The solutions are x = Dx/D and y = Dy/D. It works for any system with a unique solution.
If the determinant (D) of the system is zero, the system does not have a unique solution. It is either ‘inconsistent’ (no solution, representing parallel lines that never intersect) or ‘dependent’ (infinite solutions, representing the exact same line). This calculator will alert you if D = 0 and explain which case applies.
No, this calculator is specifically designed for linear simultaneous equations (where all variables are to the power of 1). If your equations contain squared terms (like x²), trigonometric functions, or variables multiplied together (like xy), you will need a non-linear system solver.
To find a unique solution for a system of linear equations, you generally need at least as many independent equations as you have variables. For example, 2 variables require 2 equations, 3 variables require 3 equations, and so on. If you have fewer equations than variables, the system is ‘underdetermined’ and will have infinitely many solutions.