Linear Equation Solver
Solve a system of two linear equations with two unknowns using substitution or Cramer's rule.
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How to use this calculator
x = (c₁b₂ − c₂b₁) / D; y = (a₁c₂ − a₂c₁) / D, where D = a₁b₂ − a₂b₁ (determinant).
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Write your two equations in the form ax + by = c.
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Enter the coefficients a and b, and the constant c for each equation.
- 3
The solver uses Cramer's rule to find x and y.
- 4
Verification rows confirm the solution satisfies both equations.
Frequently asked questions
What does it mean if there is no solution?
No solution means the two lines are parallel — they never intersect. This happens when the coefficient ratios are equal (a1/a2 = b1/b2) but the constant ratio differs. There is no (x, y) pair that satisfies both equations simultaneously.
What does "infinite solutions" mean?
Infinite solutions means the two equations describe the same line — one is a multiple of the other. Any point on the line satisfies both equations. This is a dependent system.
What is Cramer's rule?
Cramer's rule expresses each variable as a ratio of determinants. For a 2×2 system, x = Dx/D and y = Dy/D, where D is the determinant of the coefficient matrix, Dx replaces the x-column with constants, and Dy replaces the y-column. It is elegant but slower than Gaussian elimination for large systems.
Can I solve three equations with three unknowns?
This calculator handles 2×2 systems. For 3×3 or larger systems, use Gaussian elimination (row reduction). The principle is the same — the determinant must be non-zero for a unique solution.
Solving systems of linear equations
Three methods for solving simultaneous equations
Substitution: solve one equation for x, substitute into the other. Elimination: add/subtract multiples of equations to eliminate one variable. Cramer's rule: use determinants for a direct formula. All three give the same answer; substitution and elimination are preferred for hand calculation; Cramer's rule is elegant for 2×2 systems.
Geometric interpretation
Each linear equation represents a straight line. Two lines either intersect (unique solution), are parallel (no solution), or are the same line (infinite solutions). Solving the system finds the intersection point — the coordinates where both lines meet.
Applications of simultaneous equations
Mixture problems (combining solutions of different concentrations), break-even analysis (revenue line meets cost line), supply and demand (market equilibrium), circuit analysis (Kirchhoff's voltage law), and GPS triangulation all reduce to solving systems of linear equations.
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