System of Equations Calculator

System size Method

a₁

b₁

c₁

a₂

b₂

c₂

Form: a₁x + b₁y = c₁, a₂x + b₂y = c₂

Solution—

System Status—

System of Equations Calculator (Simultaneous Equations Solver)

Solve linear systems fast — from a single equation to 2×2 and 3×3 systems. Enter your coefficients, choose the method, and get the exact solution with steps: substitution, elimination, Cramer’s Rule, or matrix methods (Gaussian elimination).

This calculator shows:

Solutions for x, y, z (or a statement of no solution / infinitely many)
The method used with clear, step-by-step working
Determinants (D, Dx, Dy, Dz) and system status
Optional matrix form and row-reduction steps

How to Use the System of Equations Calculator

Pick a mode: single equation (ax + b = 0), 2×2 or 3×3 system.
Enter coefficients (e.g., a₁, b₁, c₁ for the first equation).
Choose a solution method (Elimination, Substitution, Cramer’s Rule, or Gaussian Elimination).
Click Solve to see the solution and steps.
If the system has no solution or infinitely many, the tool explains why.

What Is a System of Linear Equations?

A system of linear equations is a set of equations with the same variables (e.g., x and y). Solutions are the variable values that make all equations true at the same time.

Unique solution → lines/planes meet at one point
No solution → lines are parallel (inconsistent)
Infinitely many solutions → equations are multiples (dependent)

Methods Supported (With Steps)

1) Substitution

Solve one equation for a variable and substitute into the others.

2) Elimination (Addition Method)

Add/subtract multiples of equations to eliminate a variable, then back-substitute.

3) Cramer’s Rule (Determinants) — for 2×2 and 3×3

For a 2×2 system

${a1x+b1y=c1a2x+b2y=c2\begin{cases} a_1x + b_1y = c_1\\ a_2x + b_2y = c_2 \end{cases}$

Let $D = a_1b_2 – a_2b_1$ .
If $\neq 0$ :

$x=c1b2−c2b1D,y=a1c2−a2c1Dx=\frac{c_1b_2 – c_2b_1}{D},\quad y=\frac{a_1c_2 – a_2c_1}{D}$

If $D = 0$ , check $D_x, D_y$ for no solution or infinitely many.

4) Gaussian Elimination (Row Reduction)

Write the augmented matrix $[A ∣ b]$ , apply row operations to reach row-echelon form, then solve by back-substitution. The calculator can show each row step.

Example Problems (with outcomes)

Example 1 — 2×2 (Unique):

$y=1\begin{cases} 2x + y = 5\\ x – y = 1 \end{cases} \Rightarrow x=2,\; y=1$

Example 2 — 2×2 (No solution):

$solution\begin{cases} x + y = 3\\ 2x + 2y = 8 \end{cases} \Rightarrow \text{Parallel/inconsistent → no solution}$

Example 3 — 3×3 (Unique):

$z=3\begin{cases} x + y + z = 6\\ 2x – y + z = 3\\ x + 2y – z = 4 \end{cases} \Rightarrow x=1,\; y=2,\; z=3$

Example 4 — 3×3 (Infinite):
If one equation is a linear combination of the others → infinitely many solutions (dependent).

When to Use Each Method

Substitution: great for small systems when one equation is easy to isolate.
Elimination: fast and clean for many 2×2 / 3×3 sets.
Cramer’s Rule: perfect for 2×2/3×3 to show determinants and check consistency.
Gaussian Elimination: scalable and mirrors textbook/matrix methods.

Common Mistakes (and how the calculator helps)

Mixing up signs when adding/subtracting equations → steps highlight each operation.
Dividing by zero in Cramer’s Rule → the tool checks D = 0 and explains status.
Copying coefficients incorrectly → labeled inputs (a₁, b₁, c₁…) reduce errors.
Assuming every system has a solution → status reports none / infinite with reasons.

FAQ – System of Equations Solver

Q: What’s the difference between “simultaneous equations” and “system of equations”?
They’re the same thing; “simultaneous” is more common in UK usage.

Q: Can it show steps?
Yes — each method can display step-by-step working, including determinants or matrix row operations.

Q: Does it handle 3 variables?
Yes — 3×3 is supported. For larger systems, use the Gaussian elimination approach.

Q: How do I know if there’s no solution?
For 2×2 via Cramer’s Rule, if $D = 0$ but $D_x$ or $Dy≠0D_y\neq 0$ , the system is inconsistent (no solution). For 3×3, rank $(A)≠(A)\neq$ rank $([A ∣ b])$ → no solution.

Q: How do I recognise infinitely many solutions?
If $D = D x = Dy = 0$ (2×2) or rank $(A) =$ rank $([A ∣ b]) <$ number of variables, the system is dependent (infinitely many solutions).