Solve: $\frac{n}{n-4}=-\frac{32}{7}$.
Enter the exact value of $n$ as a fraction if necessary.
The problem asks us to solve the rational equation \(\frac{n}{n-4} = -\frac{32}{7}\) for the variable \(n\). First, we note that for the expression to be well-defined, the denominator cannot be zero, which means \(n - 4 eq 0\), so \(n eq 4\). The equation involves a ratio equal to another ratio, which suggests that cross-multiplication is an effective strategy to transform this into a linear equation that is easier to solve.
We start by cross-multiplying the terms of the equation:
\[7 \times n = -32 \times (n - 4).\]
Next, we distribute the \(-32\) on the right-hand side of the equation:
\[7n = -32n + (-32) \times (-4).\]
Since \(-32 \times -4 = 128\), the equation becomes:
\[7n = -32n + 128.\]
To isolate \(n\), we add \(32n\) to both sides of the equation:
\[7n + 32n = 128.\]
Combining the terms with \(n\), we get:
\[39n = 128.\]
Finally, dividing both sides by 39 gives us the value for \(n\):
\[n = \frac{128}{39}.\]
Since this value is not 4, it is a valid solution.
Rational equations appear in many fields, including physics and chemistry. For example, if you are mixing two solutions to achieve a specific concentration, you might set up an equation where the ratio of the amount of substance to the total volume is equal to a target concentration. Solving such equations helps ensure that your mixture has the exact proportions needed for an experiment or manufacturing process. Similar logic applies to balancing chemical equations or determining the rate at which tasks are completed by different workers.