Question

Solve the system of linear equations.

[x + y + z = 2]
[4x - y - 9z = -52]
[-x - y + 8z = 43]
Which of the following statements is true?
A. The unique solution to the system is (Type an exact answer in simplified form.)
B. All three equations describe the same plane. The solution is (x, y, z) .
C. The system has infinitely many solutions. The solution described in terms of variable z is (Type an exact answer in simplified form.)
D. There is no solution to the system.

Answer 1

To solve the system of linear equations, the elimination method reveals a unique solution to the system: (-2, -1, 5).

Step-by-step explanation:

We are asked to solve the following system of linear equations:

1. x + y + z = 2
2. 4x - y - 9z = -52
3. -x - y + 8z = 43

To find the solution, we can use methods such as substitution, elimination, or matrix operations. Here, we will use elimination:

1. Add equations 1 and 3 to eliminate y: 0x + 0y + 9z = 45, giving z = 5.
2. Substitute z = 5 into equation 1: x + y + 5 = 2, so x + y = -3.
3. Substitute z = 5 into equation 2: 4x - y - 45 = -52, so 4x - y = -7.
4. Now we have a new system with two equations and two unknowns:
5. x + y = -3 (A)
6. 4x - y = -7 (B)
7. Add equations A and B to eliminate y: 5x = -10, giving x = -2.
8. Substitute x = -2 into equation A: -2 + y = -3, giving y = -1.

Thus, the unique solution to the system is (-2, -1, 5).