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Let a, b, c be real numbers such that the system of linear equations 10x + 11y + 12z = a 13x + 14y + 15z = b 16x + 17y + 18z = c - 3 is consistent. Let X be the matrix X = [ a 2 1 ] [ b 1 0 ] [ c 0 -1 ] Let |X| denote the determinant of X. Find the value of (|X| + 10).

1. 5
2. 7
3. 9
4. 11

Correct answer: 7

Solution

Subtracting successive equations: (R2)-(R1): 3(x+y+z) = b-a; (R3)-(R1): 6(x+y+z) = c-3-a. Since R3-R1 = 2(R2-R1), we need c-3-a = 2(b-a) → a - 2b + c = 3 → -a + 2b - c = -3. Now expand |X| = a(1*(-1)-0*0) - 2(b*(-1)-0*c) + 1(b*0-1*c) = -a + 2b - c = -3. Therefore |X| + 10 = 7.

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