Question

Classify the system and identify the number of solutions.

2x + y + 5z = 4
3x − 2y + 2z = 2
5x − 8y − 4z = 1

Answer 1

inconsistent; none

Answer 2

No solutions

Explanation:

We need to eliminate one of the variables to solve the system. To do so, let's multiply the first equation by 8, the second equation by 4

• 2x + y + 5z = 4 (1) x 8
• 3x − 2y + 2z = 2 (2) x 4
• 5x − 8y − 4z = 1 (3)

The system becomes:

• 16x + 8y + 40z = 32 (1)
• 12x - 8y + 8z = 8 (2)
• 5x - 8y - 4z = 1 (3)

Add (1) and (2), subtract (3) from (2) to eliminate y

• 28x + 48z = 40 (4)
• 7x + 12z = 7 (5)

Then multiply the equation (5) by 4 to compare with(4)

• 28x + 48z = 28

Comparing the equations (4) and (5) we see it has no solutions as both variables have same coefficient but with different sum, 28≠40