- Distribute the multipliers 3 and -2 into their respective parentheses to expand the expression.
- Simplify the resulting terms, noticing that \(\frac{10}{4}x\) simplifies to \(\frac{5}{2}x\).
- Group the \(x\) terms (\(\frac{21}{5}x + \frac{5}{2}x\)) and constant terms (\(12 - 3\)) to combine them.
- Calculate the final result as \(\frac{67}{10}x + 9\), matching the second option: \(\boxed{\frac{67}{10}x + 9}\).
Explanation
- Distributing and Expanding the Expression
To simplify the expression \(3\left(\frac{7}{5}x + 4\right) - 2\left(\frac{3}{2} - \frac{5}{4}x\right)\), we first apply the distributive property to each part of the expression.
Distributing the 3:
\[3 \left(\frac{7}{5}x\right) + 3(4) = \frac{21}{5}x + 12\]
Distributing the -2:
\[(-2) \left(\frac{3}{2}\right) + (-2) \left(-\frac{5}{4}x\right) = -3 + \frac{10}{4}x\]
Now we combine these results:
\[\left(\frac{21}{5}x + 12\right) - 3 + \frac{10}{4}x\]
- Combining Like Terms and Simplifying
Next, we group the terms involving \(x\) together and the constant terms together:
\[\left(\frac{21}{5}x + \frac{10}{4}x\right) + (12 - 3)\]
Simplifying the fractions, notice that \(\frac{10}{4} = \frac{5}{2}\).
So the expression for the \(x\) coefficients becomes:
\[\frac{21}{5} + \frac{5}{2}\]
To add these fractions, we find a common denominator, which is 10:
\[\frac{21 \times 2}{5 \times 2} + \frac{5 \times 5}{2 \times 5} = \frac{42}{10} + \frac{25}{10} = \frac{67}{10}\]
The constant term is \(12 - 3 = 9\).
Combining these, we get:
\[\frac{67}{10}x + 9\]
- Comparison with Given Options
Comparing the result \(\frac{67}{10}x + 9\) with the given options, we see that it matches the second option. Thus, the simplified form is \(\boxed{\frac{67}{10}x + 9}\).
Examples
In physics and engineering, simplifying expressions is a fundamental skill used to analyze systems. For instance, when calculating the total force or displacement from multiple components, you often end up with complex expressions involving variables. By distributing and grouping these components, you can represent the system more simply, making it much easier to predict how the overall system will behave as variables change over time or space. This is similar to how we simplified the algebraic expression to find a clearer, more usable form.