Quick Answer – What is a System of Linear Equations?
A system of linear equations consists of two or more linear equations involving the same set of variables. In two variables (x, y), the system a₁x + b₁y = c₁ and a₂x + b₂y = c₂ can be solved using Cramer's rule, elimination, or substitution. The solution represents the intersection point of two lines in the Cartesian plane.
Theory & Cramer's Rule
For a system of two linear equations in two unknowns:
a₁x + b₁y = c₁
a₂x + b₂y = c₂
The determinant of the coefficient matrix is:
D = a₁b₂ − a₂b₁
By Cramer's rule, if D ≠ 0, the system has a unique solution:
x = D₃/D = (c₁b₂ − c₂b₁) / D
y = Dₙ/D = (a₁c₂ − a₂c₁) / D
• If D ≠ 0 → Unique solution (lines intersect at one point).
• If D = 0 and D₃ = Dₙ = 0 → Infinitely many solutions (coincident lines).
• If D = 0 but at least one of D₃, Dₙ ≠ 0 → No solution (parallel lines).
Solved Examples
Example 1: Unique Solution
Solve: 2x + 3y = 8, 4x − y = 2
D = 2(−1) − 4(3) = −2 − 12 = −14 ≠ 0 → Unique solution.
D₃ = 8(−1) − 2(3) = −8 − 6 = −14
Dₙ = 2(2) − 4(8) = 4 − 32 = −28
x = D₃/D = −14/−14 = 1, y = Dₙ/D = −28/−14 = 2
Example 2: Infinitely Many Solutions
Solve: x + 2y = 5, 2x + 4y = 10
D = 1(4) − 2(2) = 4 − 4 = 0
D₃ = 5(4) − 10(2) = 20 − 20 = 0
Dₙ = 1(10) − 2(5) = 10 − 10 = 0
All determinants zero → Infinitely many solutions (equations are proportional).
Example 3: No Solution
Solve: 3x + 2y = 7, 6x + 4y = 10
D = 3(4) − 6(2) = 12 − 12 = 0
D₃ = 7(4) − 10(2) = 28 − 20 = 8 ≠ 0
D = 0 but D₃ ≠ 0 → No solution (parallel lines).
Frequently Asked Questions
Q1. How do I enter negative coefficients in the linear equation solver?
Type a minus sign before the number, e.g., −5. The calculator automatically handles the sign during determinant calculations. You may also use scientific notation like 1e3 for 1000.
Q2. Can I use decimal coefficients like 0.5 or 3.14?
Yes. Enter decimals with a dot (e.g., 0.5, −2.75). Commas are automatically converted to dots for European decimal notation. Fractions like 1/2 should be entered as 0.5.
Q3. What does it mean when the determinant D equals zero?
When D = 0, the two lines are parallel. If the other determinants (D₃ and Dₙ) are also zero, the lines are coincident (infinitely many solutions). If either D₃ or Dₙ is non-zero, the lines are parallel and distinct (no solution).
Q4. Is Cramer's rule the only method used?
This calculator uses Cramer's rule because it elegantly exposes the determinant logic. However, the same results can be obtained by elimination, substitution, or matrix inversion. The steps show the determinant approach for pedagogical clarity.
Q5. Can this solve equations with more than two variables?
This tool is designed for two variables (x, y). For three or more variables, you would need a matrix solver or Gaussian elimination tool. However, Cramer's rule extends naturally to 3×3 systems, which may be added in a future version.