Certainly! Let's tackle each part of the problem methodically.
### Given:
- [tex]\( x = 2 \)[/tex]
- [tex]\( y = 3 \)[/tex]
- [tex]\( z \)[/tex] is a variable
### (a) Express [tex]\( x + y + z \)[/tex] in terms of [tex]\( z \)[/tex]
We begin by substituting the known values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[
x + y + z = 2 + 3 + z
\][/tex]
Combining the constants:
[tex]\[
x + y + z = 5 + z
\][/tex]
So, the expression in terms of [tex]\( z \)[/tex] is:
[tex]\[
x + y + z = 5 + z
\][/tex]
### (b) Express [tex]\( n \cdot x - 3 \cdot y + 42 \)[/tex] in terms of [tex]\( z \)[/tex]
Here, [tex]\( n \)[/tex] is an unspecified variable. We substitute [tex]\( x = 2 \)[/tex] and [tex]\( y = 3 \)[/tex]:
[tex]\[
n \cdot x - 3 \cdot y + 42 = n \cdot 2 - 3 \cdot 3 + 42
\][/tex]
Now, perform the arithmetic:
[tex]\[
n \cdot 2 - 9 + 42 = 2n - 9 + 42
\][/tex]
Combine the constants:
[tex]\[
2n + 33
\][/tex]
So, the expression in terms of [tex]\( z \)[/tex] (though it does not actually involve [tex]\( z \)[/tex]) is:
[tex]\[
2n + 33
\][/tex]
### Value of Expressions When [tex]\( z \)[/tex] is Given
Let's compute the values of both expressions for a specific [tex]\( z \)[/tex]. Suppose [tex]\( z = 0 \)[/tex]:
1. For [tex]\( x + y + z \)[/tex]:
[tex]\[
x + y + z = 5 + z = 5 + 0 = 5
\][/tex]
So the value is [tex]\( 5 \)[/tex].
2. For [tex]\( n \cdot x - 3 \cdot y + 42 \)[/tex]:
[tex]\[
2n + 33
\][/tex]
Since there’s no specified value for [tex]\( n \)[/tex], the value remains an expression in terms of [tex]\( n \)[/tex]:
[tex]\[
2n + 33
\][/tex]
If we assume [tex]\( n = 1 \)[/tex] for a concrete example:
[tex]\[
2 \cdot 1 + 33 = 2 + 33 = 35
\][/tex]
So for [tex]\( n = 1 \)[/tex], the value is [tex]\( 35 \)[/tex].
### Conclusion:
- The expression (a) [tex]\( x + y + z \)[/tex] in terms of [tex]\( z \)[/tex] is [tex]\( 5 + z \)[/tex].
- The expression (b) [tex]\( n \cdot x - 3 \cdot y + 42 \)[/tex] is [tex]\( 2n + 33 \)[/tex].
- For [tex]\( z = 0 \)[/tex], [tex]\( x + y + z \)[/tex] evaluates to [tex]\( 5 \)[/tex].
- For [tex]\( n = 1 \)[/tex], [tex]\( 2n + 33 \)[/tex] evaluates to [tex]\( 35 \)[/tex].
