Question

If x + y – z = 0, xy – yz – zx = –8, find the value of x² + y² + z².

Answer 1

Given: x + y – z = 0, xy – yz – zx = –8

We are asked to find the value of x² + y² + z²

Step-wise calculation:

1. From the first equation:

x + y – z = 0

⇒ z = x + y

2. Substitute z = x + y into the second equation: 

xy – y(x + y) – zx = –8 

xy – yx – y² – x(x + y) = –8 

xy – yx – y² – x² – xy = –8 

Since xy – yx = 0, 

y² – x² – xy = –8

x² + xy + y² = 8

3. We want to find x² + y² + z². 

Using z = x + y, 

x² + y² + z² = x² + y² + (x + y)²

x² + y² + z² = x² + y² + x² + 2xy + y²

x² + y² + z² = 2x² + 2y² + 2xy 

x² + y² + z² = 2(x² + y² + xy)

4. From Step 2,

x² + y² + xy = 8 

Substitute this into the above:

x² + y² + z² = 2 × 8

x² + y² + z² = 16

Therefore, the value of x² + y² + z² is 16.