Certainly! Let's solve this step-by-step.
1. Understanding the Relationship:
Since [tex]\(y\)[/tex] varies inversely as [tex]\(x\)[/tex], we can represent this relationship with the equation:
[tex]\[
y \propto \frac{1}{x}
\][/tex]
This means we can write:
[tex]\[
y = \frac{k}{x}
\][/tex]
where [tex]\(k\)[/tex] is a constant.
2. Finding the Constant [tex]\(k\)[/tex]:
We are given that [tex]\(y = 3\)[/tex] when [tex]\(x = 2\)[/tex]. We can use this information to solve for the constant [tex]\(k\)[/tex]:
[tex]\[
3 = \frac{k}{2}
\][/tex]
To find [tex]\(k\)[/tex], we multiply both sides by 2:
[tex]\[
k = 3 \cdot 2 = 6
\][/tex]
3. Using the Constant to Find [tex]\(x\)[/tex]:
Now we need to find the value of [tex]\(x\)[/tex] when [tex]\(y = 4\)[/tex]. Using the equation [tex]\(y = \frac{k}{x}\)[/tex], and substituting [tex]\(y = 4\)[/tex] and [tex]\(k = 6\)[/tex], we get:
[tex]\[
4 = \frac{6}{x}
\][/tex]
To find [tex]\(x\)[/tex], we rearrange the equation:
[tex]\[
x = \frac{6}{4}
\][/tex]
4. Simplifying the Expression:
Simplifying [tex]\(\frac{6}{4}\)[/tex] gives:
[tex]\[
x = 1.5
\][/tex]
So, the value of [tex]\(x\)[/tex] when [tex]\(y = 4\)[/tex] is:
[tex]\[
x = 1.5
\][/tex]
And the constant [tex]\(k\)[/tex] we found is:
[tex]\[
k = 6
\][/tex]
Thus, the complete solution is:
1. The constant [tex]\(k\)[/tex] is [tex]\(6\)[/tex].
2. When [tex]\(y = 4\)[/tex], the value of [tex]\(x\)[/tex] is [tex]\(1.5\)[/tex].
