Verified Answer :
- Distribute the negative sign: \(-\frac{10}{9}t + \frac{1}{8} + \frac{6}{18}t - \frac{3}{4}\).
- Simplify the fraction \(\frac{6}{18}\) to \(\frac{1}{3}\).
- Combine the \(t\)-terms: \(-\frac{10}{9}t + \frac{3}{9}t = -\frac{7}{9}t\).
- Combine the constants: \(\frac{1}{8} - \frac{6}{8} = -\frac{5}{8}\).
- The final simplified expression is \(\boxed{-\frac{7}{9} t - \frac{5}{8}}\).
Explanation
- Problem Analysis and Setup
The goal is to simplify the given expression:
\[\left(-\frac{10}{9} t + \frac{1}{8}\right) - \left(-\frac{6}{18} t + \frac{3}{4}\right)\]
This expression involves subtracting two binomials in terms of \(t\). To simplify it, we will distribute the subtraction (the negative sign) across the second parenthesis, combine the like terms involving \(t\), and combine the constant terms.
- Distributing and Simplifying Terms
First, we distribute the negative sign into the second set of parentheses:
\[-\frac{10}{9} t + \frac{1}{8} + \frac{6}{18} t - \frac{3}{4}\]
Next, we simplify the fraction \(\frac{6}{18}\). Since both 6 and 18 are divisible by 6, we have:
\[\frac{6}{18} = \frac{1}{3}\]
Substituting this back into the expression gives:
\[-\frac{10}{9} t + \frac{1}{8} + \frac{1}{3} t - \frac{3}{4}\]
- Combining Like Terms
To combine the \(t\)-terms, we group them together:
\[\left(-\frac{10}{9} t + \frac{1}{3} t\right) + \left(\frac{1}{8} - \frac{3}{4}\right)\]
For the \(t\)-terms, we need a common denominator, which is 9:
\[-\frac{10}{9} t + \frac{1 \times 3}{3 \times 3} t = -\frac{10}{9} t + \frac{3}{9} t = -\frac{7}{9} t\]
For the constants, we need a common denominator, which is 8:
\[\frac{1}{8} - \frac{3 \times 2}{4 \times 2} = \frac{1}{8} - \frac{6}{8} = -\frac{5}{8}\]
Combining these results, the simplified expression is:
\[-\frac{7}{9} t - \frac{5}{8}\]
- Conclusion
After combining all terms, the simplified expression is:
\[\boxed{-\frac{7}{9} t - \frac{5}{8}}\]
Examples
In many real-world scenarios, such as managing a household budget or balancing time in a schedule, you might have fluctuating values representing different costs or time commitments over a period (the variable \(t\)). By simplifying complex expressions into a single linear form \(at + b\), you can easily calculate the net effect of changes. For example, if \(t\) represents the number of hours you work, this expression could represent your net hourly earnings minus overhead expenses, helping you quickly determine your actual profit for any given amount of work hours.