Equation Solver
Solve linear equations, simultaneous 2×2 and 3×3 systems, and linear inequalities. Step-by-step solutions using Gaussian elimination. Extended: 3×3 systems. Professional: 4×4 systems with full row operations.
Quick Calculator Get a fast estimate
Solution
x = 3.000000
Verification
3×3.0000 + -9 = 0.000000
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How to Use the Equation Solver
For a linear equation ax+b=0, enter the coefficients a and b. For simultaneous equations, enter the coefficients of x, y, and the right-hand side for each equation.
The Extended Calculator adds 3×3 simultaneous systems and inequality solving. The Professional Calculator handles 3×3 and 4×4 systems with step-by-step Gaussian elimination.
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Extended Calculator More options, charts, and scenario comparison
Solve: ax + b = 0
Solution
x = 3
Verification
3×3 + -9 = 0
Rearranged
x = −-9/3 = −(-9)/3
Methods for Solving Linear Systems
Linear equation: ax + b = 0 → x = −b/a
2×2 system (Cramer's rule):
a₁x + b₁y = c₁
a₂x + b₂y = c₂
det = a₁b₂ − a₂b₁
x = (c₁b₂ − c₂b₁)/det
y = (a₁c₂ − a₂c₁)/det
Inequality rule: multiplying by negative flips sign
ax > b → x > b/a (if a > 0)
ax > b → x < b/a (if a < 0)
Gaussian Elimination Row Operations
| Operation | Symbol | Purpose |
|---|---|---|
| Row swap | Rᵢ ↔ Rⱼ | Move pivot into position |
| Scale row | Rᵢ × k | Make pivot = 1 |
| Row addition | Rᵢ − k·Rⱼ | Eliminate variable |
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Professional Calculator Complete parameters, sensitivity analysis, and detailed breakdown
System size:
Enter coefficients [col 1..3] and RHS (last column)
x
y
z
= RHS
Eq 1
Eq 2
Eq 3
Unique Solution
x = 5, y = 3, z = -2
Rank
3
Method
Gauss-Jordan Elimination
Row operations (7 steps)
- R3 − 2·R1
- R2 ÷ 2
- R1 − 1·R2
- R3 − 3·R2
- R3 ÷ -10.5
- R1 − -1.5·R3
- R2 − 2.5·R3
Frequently Asked Questions
x = −b/a. Move b to the right: ax = −b, then divide by a. Example: 5x+15=0 → 5x=−15 → x=−3. If a=0, either no solution or infinitely many.
Solve one equation for one variable, substitute into the other. Example: x+y=10 and 2x−y=2. From first: y=10−x. Substituting: 2x−(10−x)=2 → 3x=12 → x=4, y=6.
The system is either inconsistent (no solution — contradictory equations like x+y=3 and x+y=5) or dependent (infinite solutions — equations describe the same line).
A systematic method to solve n equations in n unknowns. Represent the system as an augmented matrix, then use row operations to reduce it to identity form, reading off the solution directly.
The inequality sign reverses when multiplying or dividing by a negative number. −3x < 12 → x > −4. This also applies when taking reciprocals of both sides.