Question

Solve the system of equations
4x - 4y + 4z = -4
4x+y-2z=5
-3х - 3y - 4z = -16

Answer 1

To solve the given system of equations, use the method of elimination or substitution. Multiply equations to make the coefficients of x the same. Add or subtract the equations to eliminate a variable. Solve for one variable, then substitute back to find the other variables. The solution to the system is x = 5, y = 27, and z = 21.

Step-by-step explanation:

To solve the given system of equations:

4x - 4y + 4z = -4

4x + y - 2z = 5

-3x - 3y - 4z = -16

1. We can use the method of elimination or substitution to solve the system of equations.
2. Let's use the method of elimination. Multiply the second equation by 4 to make the coefficients of x in the first and second equations the same. We get:
3. 16x + 4y - 8z = 20
4. Now, add the first equation and the modified second equation:
5. 20x - 4z = 16
6. Multiply the third equation by 4 to make the coefficients of x in the first and third equations the same. We get:
7. -12x - 12y - 16z = -64
8. Now, add the first equation and the modified third equation:
9. 20x - 8z = -68
10. We now have a new system of equations:
11. 20x - 4z = 16
12. 20x - 8z = -68
13. Subtract the second equation from the first equation:
14. 0x + 4z = 84
15. Divide both sides by 4:
16. z = 21
17. Now, substitute the value of z = 21 back into any of the original equations to find the values of x and y. Let's substitute it into the second equation:
18. 4x + y - 2(21) = 5
19. 4x + y - 42 = 5
20. Combine like terms:
21. 4x + y = 5 + 42
22. 4x + y = 47
23. We can use the value of z to find the value of y. Substitute z = 21 into the first equation:
24. 4x - 4y + 4(21) = -4
25. 4x - 4y + 84 = -4
26. Combine like terms:
27. 4x - 4y = -4 - 84
28. 4x - 4y = -88
29. Multiply the second equation by 4:
30. 4(4x + y) = 4(47)
31. 16x + 4y = 188
32. We now have a new system of equations:
33. 4x - 4y = -88
34. 16x + 4y = 188
35. Add the two equations:
36. 4x - 4y + 16x + 4y = -88 + 188
37. Combine like terms:
38. 20x = 100
39. Divide both sides by 20:
40. x = 5

Substitute the value of x = 5 back into any of the original equations to find the value of y. Let's substitute it into the second equation:

4(5) + y - 2z = 5

20 + y - 2z = 5

Combine like terms:

y - 2z = 5 - 20

y - 2z = -15

Substitute the value of z = 21 back into the equation above:

y - 2(21) = -15

y - 42 = -15

Combine like terms:

y = -15 + 42

y = 27

The solution to the system of equations is x = 5, y = 27, and z = 21.