Answered

For the inverse variation equation $x y=k$, what is the constant of variation, $k$, when $x=-3$ and $y=-2$ ?


A-6
B$-\frac{2}{3}$
C$\frac{3}{2}$
D6

Verified Answer :

  • The inverse variation equation is given by \(xy = k\).
  • Substitute the values \(x = -3\) and \(y = -2\) into the equation to get \((-3) \times (-2) = k\).
  • Multiply the values to find \(k = 6\).
  • The correct answer is \(\boxed{6}\).

Explanation

  1. Analyzing the Problem

An inverse variation is defined by the relationship \(xy = k\), where \(k\) is the constant of variation. In this problem, we are given the specific values \(x = -3\) and \(y = -2\) and we need to determine the value of \(k\).

  1. Substituting Values into the Equation

To find the constant of variation \(k\), we substitute the given values of \(x\) and \(y\) directly into the inverse variation equation \(xy = k\).
Substituting \(x = -3\) and \(y = -2\), we get:
\[(-3) \times (-2) = k\]

  1. Solving for k and Selecting the Correct Option

Now, we perform the multiplication to solve for \(k\):
\[(-3) \times (-2) = 6\]
Therefore, the constant of variation is \(k = 6\). Comparing this result with the given choices:

  • \(-6\)
  • \(-\frac{2}{3}\)
  • \(\frac{3}{2}\)
  • \(6\)

The correct option is \(6\).

Examples

Inverse variation is a fundamental concept in physics and economics. For instance, Boyle's Law states that for a fixed amount of gas at a constant temperature, the pressure \(P\) and volume \(V\) are inversely proportional, meaning \(PV = k\). If you compress a gas into a smaller container (decreasing volume), the pressure must increase proportionally to maintain the constant \(k\). Understanding these relationships helps scientists and engineers predict how systems behave when one variable is changed.

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