Question

Discuss the consistency of the system of equations $2x+3y+4z=11\\x+5y+7z=15\\3x+11y+13z=25$

Answer 1

Answer:

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.

Theoretical explanation:

A system of two linear equations can have one solution, an infinite number of solutions, or no solution. Systems of equations can be classified by the number of solutions.

If a system has at least one solution, it is said to be consistent .

If a consistent system has exactly one solution, it is independent .

If a consistent system has an infinite number of solutions, it is dependent . When you graph the equations, both equations represent the same line.

If a system has no solution, it is said to be inconsistent . The graphs of the lines do not intersect, so the graphs are parallel and there is no solution.

Answer 2

Answer:

The system of the given equations is consistent.

Step-by-step explanation:

The given equations are:

$2x+3y+4z=11$  ——- (1)

$x+5y+7z=15$   ——- (2)

$3x+11y+13z=25$  ——- (3)

Solving equations (1) & (2)

$2x+3y+4z=11$

$x+5y+7z=15$  *2

$2x+3y+4z=11$

$2x+10y+14z=30$

Solving

$7y+10z=19$

Now, Solving equations (2) & (3)

$x+5y+7z=15$

Multiplying by 3

$3x+11y+13z=25$

$3x+15y+21z=45$

$3x+11y+13z=25$

Solving, $4y+8z = 20$

$y+2z=5$       ———- (5)

Solving equations (4) and (5)

$7y+10z=19$

$y+2z=5$

Multiply by 5

$7y+10z=19$

$5y+10z=25$'

Solving $2y=-6$

$y=3$

Using in 1 and 2

$7x-z=10$

Solving equations (1) & (3)

$13x+5z=46$

Solving equations (6) & (7)

$7x-z=10$

Multiply by 5

$13x+5z=46$

Now

$35x-5z=50$

$13x+5z=46$

$48x=96$

$x=2$

From equation (1)

$2x+3y+4z=11$

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

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

$4-9+4z=11$

$4z=16$

$z = 4$

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

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

#SPJ3