Question

Solve the following system of equations for all three variables: a)3x−6y−z=3 b)4x+4y+z=−1 c)x−2y−z=−1

Answer 1

By eliminating z through combination of equations and then solving for y and x, we find that the solutions for the system of equations are x = 0, y = -1, and z = 3.

Step-by-step explanation:

To solve the simultaneous equations for the three unknowns (x, y, and z), we can use the method of substitution or elimination. The given equations are:

1. 3x - 6y - z = 3
2. 4x + 4y + z = -1
3. x - 2y - z = -1

First, we can sum equations (1) and (2) to eliminate z:

3x - 6y - z + 4x + 4y + z = 3 - 1

7x - 2y = 2 ...(4)

Then, we can sum equations (2) and (3) to also eliminate z:

4x + 4y + z + x - 2y - z = -1 - 1

5x + 2y = -2 ...(5)

Now, multiplying equation (5) by 3 and adding it to equation (4):

21x + 6y + 7x - 2y = 6 - 6

28x = 0

x = 0

Substituting x = 0 in equation (5):

5(0) + 2y = -2

2y = -2

y = -1

Finally, we substitute x = 0 and y = -1 into equation (1) to find z:

3(0) - 6(-1) - z = 3

6 - z = 3

z = 6 - 3

z = 3

Therefore, the solutions for the equations are x = 0, y = -1, and z = 3.

Answer 2

The solution to the system of equation 3x−6y−z=3 , 4x+4y+z=−1 , x−2y−z=−1 for x, y and z is 0, -1 and 3 respectively.

What is the solution to the system of equation?

3x−6y−z=3

4x+4y+z=−1

x−2y−z=−1

Subtract (3) from (1) to eliminate z

2x -4y = 4. (4)

Add (1) and (2) to eliminate z

7x - 2y = 2. (5)

2x -4y = 4. (4)

7x - 2y = 2. (5)

Multiply (5) by 2 and subtract

14x - 4y = 4

2x -4y = 4. (4)

12x = 0

x = 0/12

x = 0

Substitute x = 0 into (4)

2x -4y = 4. (4)

2(0) - 4y = 4

0 - 4y = 4

-4y = 4

y = 4/-4

y = -1

Substitute x and y into (1)

3x−6y−z=3

3(0) - 6(-1) - z = 3

0 + 6 - z = 3

- z = 3 - 6

-z = -3

z = -3/-1

z = 3

Therefore, x = 0, y = -1 and z = 3