Solve the following system of equations by using determinant method.
2x + 5y - z = -3
4x + 3y + 2z = 1
x + 2y + 3z = -5
Solution
D =
2
4
1
5
3
2
-1
2
3
= 2 × 3 × 3 + 5 × 2 × 1 + (-1) × 4 × 2 - 2 × 2 × 2 - 5 × 4 × 3 - (-1) × 3 × 1
= 18 + 10-8-8-60+3
= -45
D₁ =
-3
1
-5
5
3
2
-1
2
3
= -3 × 3 × 3 + 5 × 2 × (-5) + (-1) × 1 × 2 - (-3) × 2 × 2 - 5 × 1 × 3 - (-1) × 3 × (-5)
= -27-50-2+12-15-15
= -97
D₂ =
2
4
1
-3
1
-5
-1
2
3
= 2 × 1 × 3 + (-3) × 2 × 1 + (-1) × 4 × (-5) - 2 × 2 × (-5) - (-3) × 4 × 3 - (-1) × 1 × 1
= 6-6 + 20+20+36+1
= 77
D₃ =
2
4
1
5
3
2
-3
1
-5
= 2 × 3 × (-5) + 5 × 1 × 1 + (-3) × 4 × 2 - 2 × 1 × 2 - 5 × 4 × (-5) - (-3) × 3 × 1
= -30 + 5-24-4+100+9
= 56
D =
2
4
1
5
3
2
-1
2
3
= 2 × 3 × 3 + 5 × 2 × 1 + (-1) × 4 × 2 - 2 × 2 × 2 - 5 × 4 × 3 - (-1) × 3 × 1
= 18 + 10-8-8-60+3
= -45
D₁ =
-3
1
-5
5
3
2
-1
2
3
= -3 × 3 × 3 + 5 × 2 × (-5) + (-1) × 1 × 2 - (-3) × 2 × 2 - 5 × 1 × 3 - (-1) × 3 × (-5)
= -27-50-2+12-15-15
= -97
D₂ =
2
4
1
-3
1
-5
-1
2
3
= 2 × 1 × 3 + (-3) × 2 × 1 + (-1) × 4 × (-5) - 2 × 2 × (-5) - (-3) × 4 × 3 - (-1) × 1 × 1
= 6-6 + 20+20+36+1
= 77
D₃ =
2
4
1
5
3
2
-3
1
-5
= 2 × 3 × (-5) + 5 × 1 × 1 + (-3) × 4 × 2 - 2 × 1 × 2 - 5 × 4 × (-5) - (-3) × 3 × 1
= -30 + 5-24-4+100+9
= 56
x =
D₁
D =
-97
-45 = 2.156
y =
D₂
D =
77
-45 = -1.711
z =
D₃
D =
56
-45 = -1.244
D₁
D =
-97
-45 = 2.156
y =
D₂
D =
77
-45 = -1.711
z =
D₃
D =
56
-45 = -1.244
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