What is the solution to the equation?
$\frac{9}{-3 x}=\frac{12}{x+10}$
We are asked to solve the rational equation \(\frac{9}{-3x} = \frac{12}{x+10}\) for the variable \(x\). First, identify the constraints on the variable \(x\). Since denominators cannot be zero, we must have \(-3x eq 0\) and \(x + 10 eq 0\). This implies \(x eq 0\) and \(x eq -10\).
To simplify the left side of the equation, notice that \(9\) divided by \(-3\) is \(-3\). Thus, we can rewrite the equation as \(\frac{-3}{x} = \frac{12}{x+10}\). This step simplifies the expression and makes it easier to work with.
To solve the proportion, perform cross-multiplication: multiply the numerator of one side by the denominator of the other. This yields:
\[-3(x + 10) = 12x\]
Distribute the \(-3\) across the terms inside the parentheses:
\[-3x - 30 = 12x\]
Now, isolate \(x\) by adding \(3x\) to both sides of the equation:
\[-30 = 12x + 3x\]
\[-30 = 15x\]
Divide both sides by \(15\) to solve for \(x\):
\[x = \frac{-30}{15}\]
\[x = -2\]
Finally, check the result against our initial constraints. Since \(x = -2\) is not equal to \(0\) or \(-10\), the solution is valid. We verify by substituting \(x = -2\) back into the original equation: \(\frac{9}{-3(-2)} = \frac{9}{6} = 1.5\) and \(\frac{12}{-2+10} = \frac{12}{8} = 1.5\). Both sides are equal, confirming the result is \(\boxed{-2}\).
In real life, rational equations like this are used in physics and engineering to relate variables that have an inverse relationship. For example, in optics, the thin lens equation \(\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\) involves similar principles where you need to solve for an unknown distance based on known focal points. Mastering these equations allows you to calculate magnification, timing, or proportional scaling in various technical tasks.