Solve each system of equations by using elimination

Question

A) 7x-5y+4z-9=0

-x+3y+5z-2=0

5x+y+14z-20=0

b) 7x-5y+4z-9=0

-5x+4y+z-3=0

x+2y-z+1=0

c) -3x+2y-z+10=0

6x-4y+2z-10=0

5x+7y+9z-14=0

d) 5x-9y+2z-3=0

4x+2y-3z+4=0

9x-7y-z+1=0

Answer 1

A)

7x-5y+4z-9=0 --------- (1)

-x+3y+5z-2=0 --------- (2)

5x+y+14z-20=0       --------- (3)

Eliminate x from (1) and (2) equations:

(1) + 7(2):

7(2): 7(-x+3y+5z-2)=0

-7x + 21y + 35z - 14 = 0

(1) + 7(2):

7x - 5y + 4z - 9=0

-7x + 21y + 35z - 14 = 0

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       16 y + 39z  - 23 = 0     --------- (4)

Eliminate x from (2) and (3) equations:

5(2) + (3):

5(2): 5(-x+3y+5z-2)=0

-5x + 15y + 25z - 10 = 0

5(2) + (3):

-5x + 15y + 25z - 10 = 0

5x + y +14z -20=0

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       16 y + 39z  - 30= 0     --------- (5)

No solution for these equations.

Answer 2

b)

7x-5y+4z-9=0 --------- (1)

-5x+4y+z-3=0 --------- (2)

x+2y-z+1=0       --------- (3)

Eliminate x from (1) and (2) equations:

5(1) + 7(2):

5(1) :5(7x-5y+4z-9) = 0

35x -25y + 20z - 45 = 0

7(2): 7(-5x+4y+z-3)=0

-35x + 28y + 7z - 21 = 0

5(1) + 7(2):

35x -25y + 20z - 45 = 0

-35x + 28y + 7z - 21 = 0

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       3 y + 27z  - 66 = 0  

y + 9z  - 22 = 0 --------- (4)

Eliminate x from (2) and (3) equations:

(2) + 5(3):

5(3): 5(x+2y-z+1)=0

5x + 10y - 5z + 5 = 0

(2) + 5(3):

-5x + 4y +z -3 =0 

5x + 10y - 5z + 5 =0

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       14 y - 4z  + 2= 0     --------- (5)

14 (4)- (5):

     14y + 126z  - 308 = 0

(-) 14 y - 4z       + 2  = 0  

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             130 z - 310 =0

                  z =310/130 = 2.385

Substitute 2.385 for z in equation (5):

14 y - 4z  + 2= 0 

14 y - 4(2.385)  + 2= 0 

14 y =7.5385

y = 0.5385

Substitute 0.5385 for y,2.385 for z in equation (3):

x+2y-z+1=0 

x+2(0.5385 )-2.385+1=0 

x =0.3081   

The solution are   x =0.3081 ,y = 0.5385, and z = 2.385.

 

Answer 3

c)

-3x+2y-z+10=0  --------- (1)

6x-4y+2z-10=0  --------- (2)

5x+7y+9z-14=0       --------- (3)

Eliminate x from (1) and (2) equations:

2(1) + (2):

2(1) :2(-3x+2y-z+10) = 0

-6x +4y - 2z + 20 = 0

2(1) + (2):

-6x +4y - 2z + 20 = 0

6x -4y +2z -10 =0

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Equations (1) and (2) are parallel

No solution 

 

Answer 4

d)

5x-9y+2z-3=0  --------- (1)

4x+2y-3z+4=0  --------- (2)

9x-7y-z+1=0       --------- (3)

Eliminate x from (1) and (2) equations:

5(2) - 4(1):

5(2) :5(4x+2y-3z+4) = 0

20x +10y - 15z + 20 = 0

4(1): 4(5x-9y+2z-3)=0

20x - 36y + 8z - 12 = 0

5(2) - 4(1):

     20x +10y - 15z + 20 = 0

(-) 20x - 36y + 8z - 12 = 0

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             46 y - 23z  - 32 = 0   --------- (4)

Eliminate x from (2) and (3) equations:

9(2) - 4(3):

9(2): 9(4x+2y-3z+4)=0

36x + 18y - 27z + 36 = 0

4(3):4(9x-7y-z+1)=0

36x - 28y - 4z + 4 = 0   

9(2) - 4(3):

    36x + 18y - 27z + 36 = 0

(-) 36x - 28y - 4z + 4 = 0  

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       46 y - 23z  + 32= 0     --------- (5)

(4)- (5):

     46 y - 23z  - 32 = 0

(-) 46 y - 23z  + 32= 0

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No solution for these equations.