x+2y+5z=-1

2x-y+z=z
3x+4y-4z=14

solve the system using linear combination method.

Are you sure Eq2 = Z?

To solve the system of linear equations using the linear combination method, we need to eliminate one variable at a time from the equations. Here's how you can do it step by step:

Step 1: Rearrange the equations to align the variables and constants.

Equation 1: x + 2y + 5z = -1
Equation 2: 2x - y + z = z
Equation 3: 3x + 4y - 4z = 14

Step 2: Simplify equation 2 by canceling out z on both sides.

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

Now we have:

Equation 1: x + 2y + 5z = -1
Equation 2: 2x - y = 0
Equation 3: 3x + 4y - 4z = 14

Step 3: Multiply equation 2 by 2 and add it to equation 1 to eliminate the x variable.

2(2x - y) = 0
4x - 2y = 0

Now we have:

Equation 1: x + 2y + 5z = -1
Equation 2: 4x - 2y = 0
Equation 3: 3x + 4y - 4z = 14

x + 2y + 5z = -1
4x - 2y = 0
----------------
5x + 5z = -1 (equation 4)

Step 4: Multiply equation 2 by 3 and add it to equation 3 to eliminate the x variable.

3(4x - 2y) = 0
12x - 6y = 0

Now we have:

Equation 1: x + 2y + 5z = -1
Equation 2: 12x - 6y = 0
Equation 3: 3x + 4y - 4z = 14

12x - 6y = 0
3x + 4y - 4z = 14
---------------
15x - 4z = 14 (equation 5)

Step 5: Now we have a system of two equations with two variables (equations 4 and 5). Solve this system using any method you prefer, such as substitution, elimination or matrices.

Equation 4: 5x + 5z = -1
Equation 5: 15x - 4z = 14

Let's solve the system using elimination method:

Multiply equation 4 by 3 and equation 5 by -1:

15x + 15z = -3
-15x + 4z = -14
----------------
19z = -17

Divide both sides of equation 19z = -17 by 19:

z = -17/19

Substitute z = -17/19 back into equation 4:

5x + 5(-17/19) = -1
5x - 85/19 = -1

Now, solve for x:

5x = -1 + 85/19
5x = -19/19 + 85/19
5x = 66/19
x = 66/19 * 1/5
x = 66/95

Substitute the values of x and z into equation 5:

15x - 4z = 14
15(66/95) - 4(-17/19) = 14
990/95 + 68/19 = 14
990/95 + 340/95 = 14
1330/95 = 14

Simplify further:

14 = 14

Both sides are equal, which means the solution is valid.

Therefore, the solution to the given system of equations is:
x = 66/95, y = unknown, z = -17/19.