Cramer's Rule
\(x = \frac{D_x}{D}, y = \frac{D_y}{D}\) where \(D \neq 0\)
Yhtälöryhmälaskin
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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
\(D = 0\) and different ratios → No solution
Coincident Lines
\(D = 0\) and equal ratios → Infinite solutions