To solve this problem, we need to find the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], [tex]\( c \)[/tex], [tex]\( d \)[/tex], and [tex]\( e \)[/tex] that make the two relations inverses of each other. Two relations are inverses if the pair [tex]\( (x, y) \)[/tex] from one relation matches the pair [tex]\( (y, x) \)[/tex] in the other.
We will proceed by comparing corresponding pairs:
1. Finding [tex]\( a \)[/tex]:
- From the first table: [tex]\( (-3.8, -3.1) \)[/tex]
- From the second table, we need to find [tex]\( a \)[/tex] such that [tex]\( (-3.1, a) \)[/tex] matches with [tex]\( (-3.8, -3.1) \)[/tex].
- Therefore, [tex]\( a = -3.8 \)[/tex].
2. Finding [tex]\( b \)[/tex]:
- From the first table: [tex]\( (b, 3.2) \)[/tex]
- From the second table, we need to find [tex]\( b \)[/tex] such that [tex]\( (3.2, -2.6) \)[/tex] matches with [tex]\( (b, 3.2) \)[/tex].
- Therefore, [tex]\( b = -2.6 \)[/tex].
3. Finding [tex]\( c \)[/tex]:
- From the first table: [tex]\( (-1.4, c) \)[/tex]
- From the second table, we need to find [tex]\( c \)[/tex] such that [tex]\( (1.7, -1.4) \)[/tex] matches with [tex]\( (-1.4, c) \)[/tex].
- Therefore, [tex]\( c = 1.7 \)[/tex].
4. Finding [tex]\( d \)[/tex]:
- From the first table: [tex]\( (-0.2, 4.4) \)[/tex]
- From the second table, we need to find [tex]\( d \)[/tex] such that [tex]\( (d, -0.2) \)[/tex] matches with [tex]\( (-0.2, 4.4) \)[/tex].
- Therefore, [tex]\( d = 4.4 \)[/tex].
5. Finding [tex]\( e \)[/tex]:
- From the first table: [tex]\( (1.0, 5.0) \)[/tex]
- From the second table, we need to find [tex]\( e \)[/tex] such that [tex]\( (5.0, e) \)[/tex] matches with [tex]\( (1.0, 5.0) \)[/tex].
- Therefore, [tex]\( e = 1.0 \)[/tex].
Thus, the values that make the relations inverses of each other are:
[tex]\[
\begin{array}{l}
a = -3.8 \\
b = -2.6 \\
c = 1.7 \\
d = 4.4 \\
e = 1.0 \\
\end{array}
\][/tex]
