To solve the equation \(-\frac{5}{6}t = \frac{9}{4}\) using the multiplication property of equality, we first want to isolate \(t\).
The multiplication property of equality states that if you multiply both sides of an equation by the same non-zero number, the two sides remain equal. In this case, we can multiply both sides by the reciprocal of \(-\frac{5}{6}\), which is \(-\frac{6}{5}\).
Step 1: Multiply both sides by \(-\frac{6}{5}\)
\[ -\frac{6}{5} \cdot \left(-\frac{5}{6}t\right) = -\frac{6}{5} \cdot \frac{9}{4} \]
The left side simplifies to:
\[ t \]
For the right side:
\[ -\frac{6}{5} \cdot \frac{9}{4} = -\frac{54}{20} \]
We can simplify \(-\frac{54}{20}\):
\[ -\frac{54 \div 2}{20 \div 2} = -\frac{27}{10} \]
Thus, we find:
\[ t = -\frac{27}{10} \]
Step 2: Check the solution
Substituting \(t = -\frac{27}{10}\) back into the original equation to verify:
\[ -\frac{5}{6}t = \frac{9}{4} \]
Substituting \(t\):
\[ -\frac{5}{6} \left(-\frac{27}{10}\right) = \frac{9}{4} \]
Calculating the left side:
\[ \frac{5 \cdot 27}{6 \cdot 10} = \frac{135}{60} \]
Now simplify \(\frac{135}{60}\):
\[ \frac{135 \div 15}{60 \div 15} = \frac{9}{4} \]
Both sides of the equation are equal, confirming that our solution is correct.
Final Answer:
\[ t = -\frac{27}{10} \]