Solve a system of linear equations using Cramer's rule — 3 variable

Question

Using Cramer’s rule, solve the system of equations:

x + 2y + 3z = 6

2x - y + z = 3

3x + y - z = 4

(JEE Main 2021, similar pattern)

Solution — Step by Step

D = \begin{vmatrix} 1 & 2 & 3 \\ 2 & -1 & 1 \\ 3 & 1 & -1 \end{vmatrix}

Expand along the first row:

D = 1(1 - 1) - 2(-2 - 3) + 3(2 + 3)

D = 1(0) - 2(-5) + 3(5)

D = 0 + 10 + 15 = \mathbf{25}

Since $D \neq 0$ , the system has a unique solution.

D_1 = \begin{vmatrix} 6 & 2 & 3 \\ 3 & -1 & 1 \\ 4 & 1 & -1 \end{vmatrix}

D_1 = 6(1 - 1) - 2(-3 - 4) + 3(3 + 4)

D_1 = 6(0) - 2(-7) + 3(7)

D_1 = 0 + 14 + 21 = \mathbf{35}

D_2 = \begin{vmatrix} 1 & 6 & 3 \\ 2 & 3 & 1 \\ 3 & 4 & -1 \end{vmatrix}

D_2 = 1(-3 - 4) - 6(-2 - 3) + 3(8 - 9)

D_2 = -7 + 30 - 3 = \mathbf{20}

D_3 = \begin{vmatrix} 1 & 2 & 6 \\ 2 & -1 & 3 \\ 3 & 1 & 4 \end{vmatrix}

D_3 = 1(-4 - 3) - 2(8 - 9) + 6(2 + 3)

D_3 = -7 + 2 + 30 = \mathbf{25}

x = \frac{D_1}{D} = \frac{35}{25} = \frac{7}{5}

y = \frac{D_2}{D} = \frac{20}{25} = \frac{4}{5}

z = \frac{D_3}{D} = \frac{25}{25} = 1

\boxed{x = \frac{7}{5}, \quad y = \frac{4}{5}, \quad z = 1}

Why This Works

Cramer’s rule comes from the theory of determinants. Each variable equals the ratio of two determinants: the numerator determinant is formed by replacing the column of that variable’s coefficients with the column of constants, and the denominator is always $D$ (the coefficient determinant).

The rule works when $D \neq 0$ , which means the coefficient matrix is invertible and the system has a unique solution. If $D = 0$ , the system either has infinitely many solutions or no solution — Cramer’s rule cannot distinguish between these cases.

Alternative Method — Matrix inversion

Write the system as $AX = B$ , where $A$ is the coefficient matrix. Then $X = A^{-1}B$ .

For a $3 \times 3$ system, computing $A^{-1}$ involves finding the adjoint matrix and dividing by $|A|$ . Cramer’s rule is essentially doing this calculation column by column, which is often faster for a single system.

For JEE Main, Cramer’s rule for $3 \times 3$ systems is manageable if you’re fast with $3 \times 3$ determinants. Practice expanding determinants along the row/column with the most zeros — this speeds up calculation significantly. If $D = 0$ , switch to row reduction instead.

Common Mistake

The most common error: when computing $D_1$ , $D_2$ , $D_3$ , students replace the wrong column. Remember — $D_1$ replaces the first column (x-coefficients) with the constants, $D_2$ replaces the second column (y-coefficients), and $D_3$ replaces the third column (z-coefficients). A column replacement error gives completely wrong values for all three variables.

Question

Solution — Step by Step

Why This Works

Alternative Method — Matrix inversion

Common Mistake

Try These Next