Solve systems of 2 or 3 linear equations using Cramer's rule, showing determinants, solution type, and step-by-step work.
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System size
Equation 1
Equation 2
Solution
The system has a unique solution: x = 2.2, y = 1.2.
Step-by-Step Solution
System: 2x + 3y = 8
1x + -1y = 1
Coefficient determinant D = (2)(-1) - (3)(1) = -5
Dx = (8)(-1) - (3)(1) = -11
Dy = (2)(1) - (8)(1) = -6
x = Dx / D = -11 / -5 = 2.2
y = Dy / D = -6 / -5 = 1.2
Interpretation
The determinant D = -5 is non-zero, confirming a unique solution exists. The equations intersect at exactly one point.
Also in Functions
Algebra
A system of equations calculator solves two or three simultaneous linear equations for their unknown variables. It uses Cramer's rule (determinant-based method) to find exact solutions and identifies whether the system has a unique solution, infinitely many solutions, or no solution at all.
For a 2×2 system a₁x + b₁y = c₁ and a₂x + b₂y = c₂, the determinant D = a₁b₂ − a₂b₁. When D ≠ 0, the system has a unique solution: x = (c₁b₂ − c₂b₁)/D and y = (a₁c₂ − a₂c₁)/D.
If D = 0, the system is either inconsistent (no solution — parallel lines) or dependent (infinitely many solutions — the same line).
D = a₁b₂ − a₂b₁
Determinant of the coefficient matrix.
x = (c₁b₂ − c₂b₁) / D
Solution for x using Cramer's rule.
Frequently asked questions
A zero determinant means the equations are either parallel (no solution) or identical (infinitely many solutions). You cannot divide by zero in Cramer's rule, so the method signals that a unique solution does not exist.
No — Cramer's rule and the methods used here apply only to linear equations (variables to the first power with no products of variables). Non-linear systems require different techniques like substitution or numerical methods.
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