Simplify the expression:
[tex]\[ \left(-\frac{1}{5}r - 4 - \frac{2}{3}r\right) - \left(-\frac{4}{5}r + 9\right) \][/tex]

A. [tex]\(-\frac{25}{15}r - 5\)[/tex]

B. [tex]\(\frac{5}{7}r + (-13)\)[/tex]

C. [tex]\(-\frac{1}{15}r - 5\)[/tex]

D. [tex]\(-\frac{1}{15}r + (-13)\)[/tex]



Answer :

To simplify the expression [tex]\(\left(-\frac{1}{5} r - 4 - \frac{2}{3} r\right) - \left(-\frac{4}{5} r + 9\right)\)[/tex], let’s follow these steps methodically:

1. Distribute and Combine Like Terms:
- First, distribute the subtraction over the second group of terms:
[tex]\[ \left(-\frac{1}{5} r - 4 - \frac{2}{3} r\right) - \left(-\frac{4}{5} r + 9\right) = -\frac{1}{5} r - 4 - \frac{2}{3} r - (-\frac{4}{5} r) - 9 \][/tex]

2. Simplify the Distribution:
- Notice how subtracting a negative is equivalent to adding:
[tex]\[ -\frac{1}{5} r - 4 - \frac{2}{3} r + \frac{4}{5} r - 9 \][/tex]

3. Combine Like Terms:
- Combine the terms involving [tex]\(r\)[/tex]:
[tex]\[ -\frac{1}{5} r - \frac{2}{3} r + \frac{4}{5} r \][/tex]
- Calculate the coefficients:
[tex]\[ -\frac{1}{5} + \frac{4}{5} - \frac{2}{3} \][/tex]
- First, combine [tex]\(-\frac{1}{5}\)[/tex] and [tex]\(\frac{4}{5}\)[/tex]:
[tex]\[ -\frac{1}{5} r + \frac{4}{5} r = \frac{3}{5} r \][/tex]
- Then add the remaining term [tex]\(-\frac{2}{3}\)[/tex]:
To combine these, find a common denominator, which is 15:
[tex]\[ \frac{3}{5} = \frac{9}{15} \quad \text{and} \quad -\frac{2}{3} = -\frac{10}{15} \][/tex]
[tex]\[ \frac{9}{15} r - \frac{10}{15} r = -\frac{1}{15} r \][/tex]

4. Combine the constant terms:
[tex]\[ -4 - 9 = -13 \][/tex]

5. Write the final simplified expression:
[tex]\[ -\frac{1}{15} r - 13 \][/tex]

After following these steps, we find that the simplified form of the expression is:
[tex]\[ -\frac{1}{15} r - 13 \][/tex]

In relation to the given choices, the correct answer is:
[tex]\[ -\frac{1}{15} r + (-13) \][/tex]

So, the correct option is [tex]\( \boxed{-\frac{1}{15} r + (-13)} \)[/tex].