Let's solve the problem step by step.
We start with two given equations:
[tex]\[ x^2 + y^2 + z^2 = 36 \][/tex]
[tex]\[ xy + yz + zx = 19 \][/tex]
To find the value of [tex]\(x + y + z\)[/tex], we introduce a new variable, let [tex]\(S = x + y + z\)[/tex].
One useful identity to remember is the square of a sum:
[tex]\[ S^2 = (x + y + z)^2 = x^2 + y^2 + z^2 + 2(xy + yz + zx) \][/tex]
Using the given information, we can express this identity with the known values.
Substitute [tex]\( x^2 + y^2 + z^2 = 36 \)[/tex] and [tex]\( xy + yz + zx = 19 \)[/tex] into the identity:
[tex]\[ S^2 = 36 + 2 \cdot 19 \][/tex]
[tex]\[ S^2 = 36 + 38 \][/tex]
[tex]\[ S^2 = 74 \][/tex]
To find [tex]\(S\)[/tex], we take the square root of both sides:
[tex]\[ S = \pm \sqrt{74} \][/tex]
Simplifying [tex]\(\sqrt{74}\)[/tex]:
[tex]\[ S = \pm \sqrt{2} \sqrt{37} \][/tex]
Therefore, the two possible values for [tex]\(x + y + z\)[/tex] are:
[tex]\[ x + y + z = -\sqrt{2} \sqrt{37} \quad \text{or} \quad x + y + z = \sqrt{2} \sqrt{37} \][/tex]
So, the values of [tex]\( x + y + z \)[/tex] are:
[tex]\[ -\sqrt{2}\sqrt{19 + 18} \quad \text{and} \quad \sqrt{2}\sqrt{19 + 18} \][/tex]
Hence,
[tex]\[\boxed{\left( -\sqrt{2} \sqrt{x y + x z + y z + 18}, \sqrt{2} \sqrt{x y + x z + y z + 18} \right)}\][/tex]
