# Find x,y,z 2x+3y+5z=30, x+y+z=10, 2x-5y+4z=20

Reduction method
reshown Nov 2, 2018

The system of equations are

2x + 3y + 5z  =  30 ---------------------> (1)

x + y + z  =  10 ----------------------> (2)

2x - 5y + 4z  =  20 ------------------------> (3)

Multiply Eq (2) with 5 and add to Eq (3)

Eq (2) X 5 ======>  5x + 5y + 5z  =  50

Eq (3) =========>  2x -  5y + 4z  =  20
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7x + 9z  =  70 --------------------> (4)

Multiply Eq (2) with 5 and subtract it from Eq (1)

Eq (3) =========>  2x + 3y + 5z  =  30

Eq (2) X 3 ======>  3x + 3y + 3z  =  30

(-)   (-)    (-)         (-)

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- x + 2z  =  0

x  =  2z --------------------------> (5)

Substitute x = 2z in Eq (4)

7(2z) + 9z  =  70

14z + 9z  =  70

23z  =  70

z  =  70/23

Substitute z = 70/23 in Eq (5)

x  =  2(70/23)

x  =  140/23

Substitute z = 70/23 and z = 140/23 in Eq (2)

70/23 + y + 140/23  =  10

(70 + 140)/23 + y  =  10

(210)/23 + y  =  10

y  =  10 - (210)/23

y  =  (230 - 210)/23

y  =  20/23