Question

check the consistency of the system of equation x+5y+7z=15 , 2x+10y+14z=30 , 3x+11y+13z=2 5 and solve them.
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Answer 1

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Given equations;

2x+3y+4z=11     ——- (1)

 x+5y+7z=15     ——- (2)

3x+11y+13z=25  ——- (3)

Solving equations (1) & (2)

2x+3y+4z=11      

 x+5y+7z=15      Multiply by 2

 2x+3y+4z=11

 2x+10y+14z=30

(-)  (-)    (-)      (-)

————————

   -7y-10z=-19

   7y+10z=19       ——— (4)

Solving equations (2) & (3)

     x+5y+7z=15        Multiply by 3

3x+11y+13z=25

 3x+15y+21z=45

 3x+11y+13z=25

(-)  (-)    (-)      (-)

 ———————

       4y+8z = 20

     

         y+2z=5       ———- (5)

Solving equations (4) and (5)

7y+10z=19

y+2z=5            Multiply by 5

 7y+10z=19

 5y+10z=25

(-)  (-)      (-)

——————-

  2y=-6

  y = -3

Solving equations (1) & (2)

2x+3y+4z=11         Multiply by 5

 x+5y+7z=15         Multiply by 3

   10x+15y+20z=55

     3x+15y+21z=45

   (-)  (-)    (-)      (-)

  ————————-

    7x-z=10     ———(6)

Solving equations (1) & (3)

2x+3y+4z=11                Multiply by 11

3x+11y+13z=25            Multiply by 3

   22x+33y+44z=44

     9x+33y+39z=75

   (-)  (-)     (-)     (-)

  ————————-

     13x+5z=46      ———-(7)

Solving equations (6) & (7)

7x-z=10              Multiply by 5

13x+5z=46

   35x-5z=50

   13x+5z=46

———————

    48x=96

        x = 2

From equation (1)

2x+3y+4z=11

Substituting x=2 and y=-3 in the above equation.

2(2)+3(-3)+4z=11

4-9+4z=11

4z=16

  z = 4

Therefore, the solution is (x,y,z) = (2,-3,4)

Since we have a solution, the system of the given equations is consistent.

Answer 2

Answer:

-3x-2y+4z = -15 ---------> (1)

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

4x-y+7z = 15 ---------. (3)

multiple to each side of equation (2)by 2.

4x+10y-6z = 6 ----------> (4)

To eliminate the x value subtract equation (3) from (4).

4x+10y-6z = 6

4x-y+7z = 15

(-) (+) (-) (-)

___________

11y-13z = -9 ------------> (5)

multiple to each side of equation (2) by 3 and (1) by 2 and add the equations.

-6x-4y+8z = -30

6x+15y-9z = 9

____________

11y-z = -21 ---------> (6)

to eliminate the y value subtract the equation (6) from (5).

11y-13z =-9

11y-z = -21

(-) (+) (+)

__________

-12z = 12

Divide to each side by negitive 12.

-12z/-12 = 12/-12

z =- 1

Substitute the z value in (6)

11y+1 = -21

Subtract 1 from each side.

11y+1-1 = -21-1

11y = -22

Divide to each side by 11.

y/11 = -22/11

y = -2

Substitute the z,y values in (2).

2x+5*-2-3*-1 = 3

2x-10+3 = 3

2x -7 = 3

add 7 to each side.

2x -7+7 = 7+3

2x = 10

Divide to each side by 2.

2x/2 = 10/2

x = 5

Solution of given system is x = 5, y = -2, z = -1.

Hope it helps you!