To solve for z in the system of equations given, we need to eliminate variables step by step. Let's go through the process:
1. Start by rearranging the equations to isolate z:
- Equation 1: 6w - 4z + 2y + 8x = 52
Rearrange: -4z = -6w - 2y - 8x + 52
Divide by -4: z = 1.5w + 0.5y + 2x - 13
- Equation 2: 2y - 4w - 6x - 2z = -28
Rearrange: -2z = -2y + 4w + 6x - 28
Divide by -2: z = y - 2w - 3x + 14
- Equation 3: 6z + 2x - 4w + 4y = 36
Rearrange: 6z = -2x + 4w - 4y + 36
Divide by 6: z = -0.33x + 0.67w - 0.67y + 6
- Equation 4: 4x + 4y - 2w - 2z = 20
Rearrange: -2z = -4x - 4y + 2w + 20
Divide by -2: z = 2x + 2y - w - 10
2. Now, we have z expressed in terms of the other variables in each equation:
- z = 1.5w + 0.5y + 2x - 13
- z = y - 2w - 3x + 14
- z = -0.33x + 0.67w - 0.67y + 6
- z = 2x + 2y - w - 10
3. Equate the expressions for z from the different equations:
1.5w + 0.5y + 2x - 13 = y - 2w - 3x + 14
Solve this equation to find the values of w, x, and y that satisfy the system and consequently find z.
By following these steps, you can solve for z in the system of equations provided.
