To solve the given problem regarding the linear functions [tex]\( v \)[/tex] and [tex]\( w \)[/tex], where [tex]\( v(10) = 12 \)[/tex] and [tex]\( w(10) = 9 \)[/tex], we need to determine the following:
1. [tex]\( (v + w)(10) \)[/tex]
2. [tex]\( (v - w)(10) \)[/tex]
3. [tex]\(-(vw)(10)\)[/tex]
Let’s break it down step-by-step:
### 1. Calculating [tex]\( (v + w)(10) \)[/tex]:
[tex]\( (v + w)(10) \)[/tex] represents the value obtained when we add the values of [tex]\( v(10) \)[/tex] and [tex]\( w(10) \)[/tex].
Given:
- [tex]\( v(10) = 12 \)[/tex]
- [tex]\( w(10) = 9 \)[/tex]
So,
[tex]\[ (v + w)(10) = v(10) + w(10) = 12 + 9 = 21 \][/tex]
### 2. Calculating [tex]\( (v - w)(10) \)[/tex]:
[tex]\( (v - w)(10) \)[/tex] represents the value obtained when we subtract [tex]\( w(10) \)[/tex] from [tex]\( v(10) \)[/tex].
Given:
- [tex]\( v(10) = 12 \)[/tex]
- [tex]\( w(10) = 9 \)[/tex]
So,
[tex]\[ (v - w)(10) = v(10) - w(10) = 12 - 9 = 3 \][/tex]
### 3. Calculating [tex]\(-(vw)(10)\)[/tex]:
[tex]\( -(vw)(10) \)[/tex] represents the negative value of the product of [tex]\( v(10) \)[/tex] and [tex]\( w(10) \)[/tex].
Given:
- [tex]\( v(10) = 12 \)[/tex]
- [tex]\( w(10) = 9 \)[/tex]
First, calculate the product:
[tex]\[ (vw)(10) = v(10) \cdot w(10) = 12 \cdot 9 = 108 \][/tex]
Then, take the negative:
[tex]\[ -(vw)(10) = -108 \][/tex]
### Summary of Results
- [tex]\( (v + w)(10) = 21 \)[/tex]
- [tex]\( (v - w)(10) = 3 \)[/tex]
- [tex]\( -(vw)(10) = -108 \)[/tex]
These values should be entered in the respective boxes.
