Exercise 899Linear algebra › DeterminantsHigh schoolAdded 3 Nov 2026

Cramer's rule for a 2×2 system

#Cramer's rule#linear system#determinant

Statement

Solve using Cramer's rule:
$\begin{cases} 3x + 2y = 7 \\ x - y = 1 \end{cases}$

Hint

For $AX=B$ with $\det A\neq0$ , $x_i=\det A_i/\det A$ , where $A_i$ is $A$ with column $i$ replaced by $B$ .

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Determinant.
$\det A = \begin{vmatrix} 3 & 2 \\ 1 & -1 \end{vmatrix} = -3-2=-5\neq0.$

Determinant $A_1$ .
$\det A_1 = \begin{vmatrix} 7 & 2 \\ 1 & -1 \end{vmatrix} = -7-2=-9.$

Determinant $A_2$ .
$\det A_2 = \begin{vmatrix} 3 & 7 \\ 1 & 1 \end{vmatrix} = 3-7=-4.$

Solution.
$x = \frac{-9}{-5} = \frac95, \qquad y = \frac{-4}{-5}=\frac45.$

Verification. $3\cdot9/5+2\cdot4/5=(27+8)/5=35/5=7$ and $9/5-4/5=5/5=1$ . Correct.