Question
Solve the following system of equations using Cramer's rule.
x + y - z = 6
3x - 2y + z = -5
x + 3y - 2z = 14
Answer :
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To solve the given system of equations using Cramer's rule, first write the coefficient matrix A: [ A = \begin{bmatrix} 1 & 1 & -1 \ 3 & -2 & 1 \ 1 & 3 & -2 \end{bmatrix} ] and the constant matrix B: [ B = \begin{bmatrix} 6 \ -5 \ 14 \end{bmatrix} ] The determinant of A, (|A|), is calculated using the formula for a 3×3 determinant: [ |A| = 1((-2)(-2) - (1)(3)) - 1(3(-2) - (1)(1)) + (-1)(3(3) - (-2)(1)) ] [ |A| = 1(4 - 3) - 1(-6 - 1) - 1(9 + 2) ] [ |A| = 1 - (-7) - 11 = 1 + 7 - 11 = -3 ] Next, replace the first column of A with B to get (A_x): [ A_x
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