To determine the equation of variation for [tex]\(z\)[/tex] in terms of [tex]\(x\)[/tex] and [tex]\(y\)[/tex], we start by using the given conditions and steps for variations.
### Step 1: Set Up the Variation Equation
Given that [tex]\(z\)[/tex] varies directly with [tex]\(y\)[/tex] and inversely with [tex]\(x\)[/tex], we write the relationship as:
[tex]\[
z = k \frac{y}{x}
\][/tex]
where [tex]\(k\)[/tex] is the constant of variation.
### Step 2: Substitute the Given Values
We know that when [tex]\(y = -2\)[/tex], [tex]\(x = -1\)[/tex], and [tex]\(z = -22\)[/tex], we can substitute these values into the equation to solve for [tex]\(k\)[/tex]:
[tex]\[
-22 = k \frac{-2}{-1}
\][/tex]
### Step 3: Simplify the Equation
Simplify the right-hand side of the equation:
[tex]\[
-22 = k \cdot 2
\][/tex]
### Step 4: Solve for [tex]\(k\)[/tex]
Divide both sides of the equation by 2 to solve for the constant [tex]\(k\)[/tex]:
[tex]\[
k = \frac{-22}{2} = -11
\][/tex]
### Step 5: Write the Equation of Variation
Now that we have found [tex]\(k\)[/tex], we can write the equation of variation using the relationship [tex]\(z = k \frac{y}{x}\)[/tex]:
[tex]\[
z = -11 \frac{y}{x}
\][/tex]
Therefore, the equation of variation is
[tex]\[
z = \frac{-11y}{x}
\][/tex]
