Let's solve each part step by step.
### Part (i): Calculate [tex]\( x - 2y + 3z \)[/tex]
Given:
- [tex]\( x = 4 \)[/tex]
- [tex]\( y = 3 \)[/tex]
- [tex]\( z = -2 \)[/tex]
We need to find the value of [tex]\( x - 2y + 3z \)[/tex].
Substitute the given values into the expression:
[tex]\[ x - 2y + 3z = 4 - 2(3) + 3(-2) \][/tex]
Now, evaluate each term one by one:
[tex]\[ 2(3) = 6 \][/tex]
[tex]\[ 3(-2) = -6 \][/tex]
So, the expression becomes:
[tex]\[ 4 - 6 - 6 \][/tex]
Combine the terms:
[tex]\[ 4 - 6 = -2 \][/tex]
[tex]\[ -2 - 6 = -8 \][/tex]
Thus, the value of [tex]\( x - 2y + 3z \)[/tex] is [tex]\( -8 \)[/tex].
### Part (ii): Calculate [tex]\( xy + yz + zx \)[/tex]
We now need to find the value of [tex]\( xy + yz + zx \)[/tex].
Substitute the given values into the expression:
[tex]\[ xy + yz + zx = 4(3) + 3(-2) + (-2)(4) \][/tex]
Now, evaluate each term one by one:
[tex]\[ 4(3) = 12 \][/tex]
[tex]\[ 3(-2) = -6 \][/tex]
[tex]\[ (-2)(4) = -8 \][/tex]
So, the expression becomes:
[tex]\[ 12 - 6 - 8 \][/tex]
Combine the terms:
[tex]\[ 12 - 6 = 6 \][/tex]
[tex]\[ 6 - 8 = -2 \][/tex]
Thus, the value of [tex]\( xy + yz + zx \)[/tex] is [tex]\( -2 \)[/tex].
To summarize:
1. The value of [tex]\( x - 2y + 3z \)[/tex] is [tex]\( -8 \)[/tex].
2. The value of [tex]\( xy + yz + zx \)[/tex] is [tex]\( -2 \)[/tex].
