Algebra Calculator
The Algebra Calculator calculates exact, real-time domain outputs in active voice. The Algebra Calculator solves systems of two linear equations in two unknowns (x and y) using Cramer's determinant rule, displaying the main determinant Δ, sub-determinants Δx and Δy, exact intersection coordinates (x, y), and single-variable linear reductions.
Interactive Algebra Calculator Workspace
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Algebra Calculator Input Parameters & Defaults
| Eq 1: x coefficient (a₁) | 2 | Coefficient a₁ in a₁x + b₁y = c₁. |
| Eq 1: y coefficient (b₁) | 3 | Coefficient b₁ in a₁x + b₁y = c₁. |
| Eq 1: Constant (c₁) | 13 | Right-hand constant c₁. |
| Eq 2: x coefficient (a₂) | 5 | Coefficient a₂ in a₂x + b₂y = c₂. |
| Eq 2: y coefficient (b₂) | -1 | Coefficient b₂ in a₂x + b₂y = c₂. |
| Eq 2: Constant (c₂) | 7 | Right-hand constant c₂. |
Algebra Calculator Mathematical Formula
\Delta = a_1 b_2 - a_2 b_1, \quad x = \frac{c_1 b_2 - c_2 b_1}{\Delta}, \quad y = \frac{a_1 c_2 - a_2 c_1}{\Delta}Algebra Calculator Worked Example
Solving the simultaneous system: 2x + 3y = 13 and 5x - y = 7.
- System Determinant Δ = (2)(-1) - (5)(3) = -2 - 15 = -17.
- Δx = (13)(-1) - (7)(3) = -13 - 21 = -34 -> x = -34 / -17 = 2.
- Δy = (2)(7) - (5)(13) = 14 - 65 = -51 -> y = -51 / -17 = 3.
The two lines intersect uniquely at the coordinate point (x = 2, y = 3). Substituting back confirms 2(2) + 3(3) = 13 and 5(2) - 3 = 7.
Algebra Calculator Glossary of Key Terms
Eq 1: x coefficient (a₁) (Algebra Calculator)
Coefficient a₁ in a₁x + b₁y = c₁.
Eq 1: y coefficient (b₁) (Algebra Calculator)
Coefficient b₁ in a₁x + b₁y = c₁.
Eq 1: Constant (c₁) (Algebra Calculator)
Right-hand constant c₁.
Eq 2: x coefficient (a₂) (Algebra Calculator)
Coefficient a₂ in a₂x + b₂y = c₂.
Abstract Syntax Tree (AST)
A hierarchical tree representation of a mathematical expression that enforces operator precedence without dynamic code execution.
IEEE 754 Precision Normalization
A numerical rounding method that eliminates binary floating-point representation drift in decimal arithmetic.
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Algebra Calculator Frequently Asked Questions
Q: What happens when the determinant Δ equals zero?
When Δ = 0, the two lines have identical slopes. If Δx = 0 and Δy = 0, the lines are coincident (infinitely many solutions); otherwise, the lines are parallel and never intersect (no solution).