5x-6y+4z=15, 7x+4y-3z=19, 2x+y+6z=46 solve by elimination method
Answers
1. 5x – 6y + 4z = 15
7x + 4y – 3z = 19
2x + y + 6z = 46
Hint : 1 2 3 x ,y ,z
∆ ∆ ∆ === ∆∆∆
Sol. i) Cramer’s rule :
5 64
74 3
21 6
−
∆= −
5(24 3) 6(42 6) 4(7 8)
135 288 4 419
= ++ ++ −
= + −=
1
15 6 4
19 4 3
46 1 6
−
∆= −
15(24 3) 6(114 138) 4(19 184)
405 1512 660 1917 660 1257
= ++ + + −
=+ −= −=
2
5 15 4
7 19 3
2 46 6
∆= −
5(114 138) 15(42 6) 4(322 38)
1260 720 1136 1676
= + − ++ −
= −+ =
3
5 6 15
7 4 19
2 1 46
−
∆ =
5(184 19) 6(322 38) 15(7 8)
825 1074 15 2529 15 2514x 3
419
1676 y 4 419
2514 z 6
419
∆ == = ∆
∆ == = ∆
∆ == = ∆
Solution is x = 3, y = 4, z = 6.
ii) Matrix inversion method :
Hint : 1 AdjA A
det A
− =
5 64
A 74 3
21 6
−
= −
1
1
1
4 3
A 24 3 27
1 6
7 3
B (42 6) 48 1 6
7 4
C 78 1
2 1
− = = +=
− =− =− + =−
= = − =−
2
2
2
6 4
A ( 36 4) 40 1 6
5 4
B 30 8 22
2 6
5 6
C (5 12) 17 2 1
− =− =− − − =
= = −=
− =− =− + =−
3
6 4
A 18 16 2
4 3
− = =−= −
3
5 4
B ( 15 28) 43 7 3 =− =− − − = −
3
5 6
C 20 42 62
7 4
− = =+=
Adj A =
123
123
123
A A A 27 40 2
B B B 48 22 43
C C C 1 17 62
= −
− −
Det A = ∆ = 419A 48 22 43
DetA 419
1 17 62
−
= =
− −
1
27 40 2 15
1 x A D 48 22 43 19
419
1 17 62 46
405 760 92
1 720 418 1978
419
15 323 2852
1257 3
1 1676 4
419
2514 6
−
= =−
− −
++ +
= −+ +
−− +
= =
∴ Solution is x = 3, y = 4, z = 6.
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