Simultaneous Equations Solver
Solve a system of two linear equations in two unknowns. Enter the coefficients of a₁x + b₁y = c₁ and a₂x + b₂y = c₂ — the solver returns (x, y), or flags cleanly when the lines are parallel or identical.
What is a system of equations?
A system of equations is a set of two or more equations that share the same variables. A solution makes every equation true at once. For two linear equations in x and y, each equation is a straight line and the solution is where the lines meet — one point (unique), never (parallel), or every point on the shared line (coincident).
Solving 2×2 systems, piece by piece
Each card explains one part of what the calculator does — the standard form, the two solving techniques, and the two special cases where a unique answer doesn't exist.
Standard form and geometry
Enter each equation as ax + by = c. Each row is a straight line in the plane; the solution is where the two lines meet. The determinant a₁b₂ − a₂b₁ tells you upfront whether that intersection is a single point.
Substitution method
Rearrange one equation to isolate a variable, plug that expression into the other, and you're left with a single equation in one unknown. Best when a variable already has coefficient 1 or −1.
Elimination method
Scale the two equations so one variable's coefficients match (or are exact opposites), then add or subtract to cancel that variable. This calculator uses elimination internally and prints every step.
No solution or infinitely many
If the two lines are parallel, elimination collapses to a contradiction like 0 = 5 — no solution. If they're the same line, it collapses to 0 = 0 — every point on the line is a solution. Both cases correspond to a₁b₂ − a₂b₁ = 0.
Features of this calculator
- Solves any 2×2 linear system in the form ax + by = c
- Accepts negative and decimal coefficients
- Detects parallel lines (no solution) and coincident lines (infinite solutions)
- Shows the elimination method step by step, including the check substitution
- Reports the method used and the geometric interpretation
Frequently asked questions
+Which method should I use — substitution or elimination?
Use substitution when one equation already has a variable with coefficient 1 or −1. Otherwise elimination is usually faster because you don't have to manipulate fractions.
+What is the determinant and why does it matter?
For a 2×2 system, the determinant is a₁b₂ − a₂b₁. When it's nonzero the system has a unique solution; when it's zero the lines are either parallel or identical.
+Can decimal or negative coefficients cause problems?
No. The calculator handles any real-number coefficients the same way. If the arithmetic gets messy by hand, that's when elimination tends to shine.
+How do I enter an equation like y = 2x + 1?
Rewrite it as −2x + y = 1 first, then enter a = −2, b = 1, c = 1. Any linear equation can be reshaped into the ax + by = c form.