To determine the relationship between [tex]\(y\)[/tex] and [tex]\(x\)[/tex], we start by recognizing that since [tex]\(y\)[/tex] and [tex]\(x\)[/tex] are in direct proportion, their relationship can be expressed as [tex]\(y = kx\)[/tex], where [tex]\(k\)[/tex] is a proportionality constant.
Given:
- [tex]\(x_1 = 3\)[/tex]
- [tex]\(x_2 = 8\)[/tex]
- Difference in [tex]\(y\)[/tex] values when [tex]\(x = 3\)[/tex] and [tex]\(x = 8\)[/tex] is 20.
This piece of information can be mathematically represented as:
[tex]\[ y_2 - y_1 = 20 \][/tex]
where [tex]\(y_1\)[/tex] and [tex]\(y_2\)[/tex] are the corresponding [tex]\(y\)[/tex] values for [tex]\(x_1\)[/tex] and [tex]\(x_2\)[/tex].
Since [tex]\(y\)[/tex] is directly proportional to [tex]\(x\)[/tex],
[tex]\[ y_1 = k x_1 \][/tex]
[tex]\[ y_2 = k x_2 \][/tex]
Therefore,
[tex]\[ y_2 - y_1 = k x_2 - k x_1 \][/tex]
[tex]\[ 20 = k x_2 - k x_1 \][/tex]
[tex]\[ 20 = k (x_2 - x_1) \][/tex]
Substituting the given values of [tex]\(x_1\)[/tex] and [tex]\(x_2\)[/tex]:
[tex]\[ 20 = k (8 - 3) \][/tex]
[tex]\[ 20 = k (5) \][/tex]
Solving for [tex]\(k\)[/tex]:
[tex]\[ k = \frac{20}{5} = 4.0 \][/tex]
Now that we have determined the proportionality constant [tex]\(k = 4.0\)[/tex], we can express [tex]\(y\)[/tex] in terms of [tex]\(x\)[/tex]:
[tex]\[ y = 4.0 x \][/tex]
Therefore, the equation representing [tex]\(y\)[/tex] in terms of [tex]\(x\)[/tex] is:
[tex]\[ y = 4.0 x \][/tex]
