Solving Systems of Linear Equations via Cramer's Rule
A system of two linear equations in two variables describes two straight lines in a two-dimensional Cartesian plane:
\begin{cases} a_1 x + b_1 y = c_1 \\ a_2 x + b_2 y = c_2 \end{cases}
Cramer's Rule Determinants
Gabriel Cramer published this determinant method in 1750. For a $2 \times 2$ system, define three determinants:
- Coefficient Determinant ($D$):
D = \begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix} = a_1 b_2 - a_2 b_1
- $X$-Determinant ($D_x$): Replace the $x$-column with constants $c_1, c_2$:
D_x = \begin{vmatrix} c_1 & b_1 \\ c_2 & b_2 \end{vmatrix} = c_1 b_2 - c_2 b_1
- $Y$-Determinant ($D_y$): Replace the $y$-column with constants $c_1, c_2$:
D_y = \begin{vmatrix} a_1 & c_1 \\ a_2 & c_2 \end{vmatrix} = a_1 c_2 - a_2 c_1
If $D \ne 0$, the unique intersection is given by:
x = \frac{D_x}{D}, \quad y = \frac{D_y}{D}
Geometric Classification
| System Type | Determinants | Geometry | Solutions |
|---|---|---|---|
| Independent & Consistent | $D \ne 0$ | Lines cross at one point | Unique $(x, y)$ |
| Inconsistent | $D = 0, D_x \ne 0$ | Parallel distinct lines | No solution |
| Dependent & Consistent | $D = 0, D_x = 0, D_y = 0$ | Coincident lines | Infinitely many |