Let's call the sum [tex]\(S = x + y + z\)[/tex]. To find [tex]\(S\)[/tex], we will use the given equations:
1. [tex]\(x^2 + y^2 + z^2 = 36\)[/tex]
2. [tex]\(xy + yz + zx = 19\)[/tex]
We also know that:
[tex]\[
(x + y + z)^2 = x^2 + y^2 + z^2 + 2(xy + yz + zx)
\][/tex]
We can rewrite this expression in terms of [tex]\(S\)[/tex]:
[tex]\[
S^2 = x^2 + y^2 + z^2 + 2(xy + yz + zx)
\][/tex]
Plugging in the given values from our equations:
[tex]\[
S^2 = 36 + 2 \cdot 19
\][/tex]
Calculate the value:
[tex]\[
S^2 = 36 + 38
\][/tex]
[tex]\[
S^2 = 74
\][/tex]
Taking the square root of both sides gives us:
[tex]\[
S = \sqrt{74} \quad \text{or} \quad S = -\sqrt{74}
\][/tex]
Hence, the value of [tex]\(x + y + z\)[/tex] is either [tex]\(\sqrt{74}\)[/tex] or [tex]\(-\sqrt{74}\)[/tex].
