Example: 2x2 system with Cramer

Let us see Cramer’s rule applied to a numerical system, with a final check of the solutions found.

Example — $2\times 2$ system with Cramer

$\begin{cases} 2x + 3y = -1 \\ -3x + y = 3 \end{cases}$ We compute the three determinants: $\begin{aligned} \Delta &= \begin{vmatrix} 2 & 3 \\ -3 & 1 \end{vmatrix} = 2\cdot 1 - 3\cdot(-3) = 2+9 = 11 \\[4pt] \Delta_x &= \begin{vmatrix} -1 & 3 \\ 3 & 1 \end{vmatrix} = (-1)\cdot 1 - 3\cdot 3 = -10 \\[4pt] \Delta_y &= \begin{vmatrix} 2 & -1 \\ -3 & 3 \end{vmatrix} = 2\cdot 3 - (-1)(-3) = 6-3 = 3 \end{aligned}$ Hence: $x = \frac{-10}{11} = \boxed{-\frac{10}{11}}, \qquad y = \frac{3}{11} = \boxed{\frac{3}{11}}$

Check: $2\cdot\bigl(-\frac{10}{11}\bigr) + 3\cdot\frac{3}{11} = \frac{-20+9}{11} = -1$ ✓;   $-3\cdot\bigl(-\frac{10}{11}\bigr) + \frac{3}{11} = \frac{30+3}{11} = 3$ ✓.

Connections

Topics: Linear systems
Concepts: Determinant · Cramer’s rule
Methods: Systems by Cramer
Skills: Computing · Solving systems
People: Gabriel Cramer