Cramer's Rule
\(x = \frac{D_x}{D}, y = \frac{D_y}{D}, \quad D \neq 0\)
System of Equations
Solve 2×2 systems of linear equations with Cramer's rule, step by step
Results
Enter values and click Calculate to see the result.
System of Linear Equations
A system of two linear equations can be solved using Cramer's rule, which uses determinants.
Key Concepts:
- If D ≠ 0: Unique solution (lines intersect at one point)
- If D = 0 and ratios equal: Infinite solutions (same line)
- If D = 0 and ratios differ: No solution (parallel lines)
- The determinant represents the "area" of the coefficient matrix
\(\begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases}\)
Worked Examples
Example: 2x + 3y = 8, x − y = 1
\(D = -5, x = 2.2, y = 1.2\)
Parallel Lines
\(x+y=1, \quad x+y=2\): No solution
Coincident Lines
\(x+y=1, \quad 2x+2y=2\): Infinitely many solutions