Verified Answer :

  • Start with the equation \(\frac{2}{3}n = -12\).
  • Multiply both sides by the reciprocal \(\frac{3}{2}\) to isolate the variable \(n\).
  • Calculate the result: \(n = -12 \times \frac{3}{2} = -18\).
  • The final solution is \(n = \boxed{-18}\).

Explanation

  1. Analyzing the equation

The given equation is \(\frac{2}{3}n = -12\). Our goal is to determine the value of the variable \(n\) that satisfies this equality. We observe that \(n\) is being multiplied by the fraction \(\frac{2}{3}\). To isolate \(n\), we need to perform an algebraic operation that cancels out the coefficient \(\frac{2}{3}\).

  1. Isolating the variable \(n\)

To isolate \(n\), we multiply both sides of the equation by the reciprocal of \(\frac{2}{3}\), which is \(\frac{3}{2}\). Multiplying both sides by \(\frac{3}{2}\) maintains the equality:
\[\frac{3}{2} \cdot \left(\frac{2}{3} n\right) = \frac{3}{2} \cdot (-12)\]
This simplifies the left side to \(n\):
\[n = \frac{3}{2} \cdot (-12)\]

  1. Solving for \(n\)

Now, we perform the multiplication on the right side of the equation:
\[n = \frac{3 \cdot (-12)}{2}\]
\[n = 3 \cdot (-6)\]
\[n = -18\]
Thus, the value of \(n\) that makes the original equation true is \(-18\).

Examples

Linear equations like this are fundamental in many daily tasks. For instance, if you are working on a budget and know that two-thirds of your remaining savings (\(n\)) equals \(\$12\) in debt, solving this equation helps you find your total remaining savings. Understanding these simple relationships allows you to manage resources and solve problems efficiently in personal finance, construction measurements, or basic recipe adjustments.

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