To solve for the values of [tex]\( a \)[/tex] through [tex]\( e \)[/tex] that make these two relations inverses of each other, let's proceed step by step.
### Understanding Relations and Their Inverses
Given two sets of pairs, a relation and its inverse share a specific property: if a pair [tex]\((x, y)\)[/tex] is in the first relation, then the pair [tex]\((y, x)\)[/tex] should be in the inverse relation.
### Relations Given
1. First Relation:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-3.8 & -3.1 \\
b & 3.2 \\
-1.4 & c \\
-0.2 & 4.4 \\
1.0 & 5.0 \\
\hline
\end{array}
\][/tex]
2. Second Relation (Inverse):
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-3.1 & a \\
3.2 & -2.6 \\
1.7 & -1.4 \\
d & -0.2 \\
5.0 & e \\
\hline
\end{array}
\][/tex]
### Solving for Each Variable
#### [tex]\( a \)[/tex]
It’s given that the pair [tex]\((-3.8, -3.1)\)[/tex] is in the first relation.
Thus, the inverse relation should include [tex]\((-3.1, -3.8)\)[/tex].
So, [tex]\( a = -3.8 \)[/tex].
#### [tex]\( b \)[/tex]
It’s given that the pair [tex]\(( b, 3.2 )\)[/tex] is in the first relation.
Thus, the inverse relation should include [tex]\((3.2, b )\)[/tex].
Since [tex]\((3.2, -2.6)\)[/tex] is a pair in the inverse relation, it corresponds to [tex]\( b = -2.6 \)[/tex].
#### [tex]\( c \)[/tex]
It's given that [tex]\((-1.4, c)\)[/tex] is in the first relation.
Thus, the inverse relation should include [tex]\((c, -1.4)\)[/tex].
Since [tex]\((1.7, -1.4)\)[/tex] is a pair in the inverse relation, it corresponds to [tex]\( c = 1.7 \)[/tex].
#### [tex]\( d \)[/tex]
It’s given that the pair [tex]\((-0.2, 4.4)\)[/tex] is in the first relation.
Thus, the inverse relation should include [tex]\((4.4, -0.2)\)[/tex].
Since there is no pair [tex]\( (d, 4.4) \)[/tex] included in the given relations, [tex]\( d\)[/tex] includes no value, hence [tex]\( d = None \)[/tex].
#### [tex]\( e \)[/tex]
It’s given that the pair [tex]\((1.0, 5.0)\)[/tex] is in the first relation.
Thus, the inverse relation should include [tex]\((5.0, 1.0)\)[/tex].
Since [tex]\((5.0, 1.0)\)[/tex] is a pair in the inverse relation, thus we find:
Hence, [tex]\( e = 1.0 \)[/tex].
### Summary of Values:
[tex]\[
a = -3.8 \\
b = -2.6 \\
c = 1.7 \\
d = None \\
e = 1.0
\][/tex]
Thus, the values of [tex]\( a \)[/tex] through [tex]\( e \)[/tex] that make the relations inverses of each other are:
[tex]\[
a = -3.8, \quad b = -2.6, \quad c = 1.7, \quad d = None, \quad e = 1.0
\][/tex]
