To solve the problem of finding the constant of variation when [tex]\(y\)[/tex] is directly proportional to [tex]\(x\)[/tex], let's follow these steps:
1. Understand Direct Proportionality: When we say that [tex]\(y\)[/tex] is directly proportional to [tex]\(x\)[/tex], we mean that there is a constant [tex]\(k\)[/tex] such that [tex]\(y = kx\)[/tex] for all pairs of corresponding values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
2. Identify the Relationship: We are given the pairs [tex]\((x, y)\)[/tex]:
- [tex]\((1, -4)\)[/tex]
- [tex]\((2, -8)\)[/tex]
- [tex]\((3, -12)\)[/tex]
- [tex]\((4, -16)\)[/tex]
- [tex]\((5, -20)\)[/tex]
3. Determine the Constant of Variation [tex]\(k\)[/tex]: To find [tex]\(k\)[/tex], we can use any pair of values; the relationship [tex]\(y = kx\)[/tex] holds for all data points.
4. Using the First Pair [tex]\((1, -4)\)[/tex]:
[tex]\[
y = kx
\][/tex]
[tex]\[
-4 = k \cdot 1
\][/tex]
Solving for [tex]\(k\)[/tex]:
[tex]\[
k = -4
\][/tex]
Therefore, the constant of variation [tex]\(k\)[/tex] is [tex]\(-4\)[/tex].
