Question: Subtract the following:
1) $\frac{5}{8}a - \frac{2}{8}a$
2) $\frac{7}{3}x - \frac{2}{4}y$
3) $\frac{6}{x + 4} - \frac{2}{x - 1}$
4) $\frac{4}{x - 2} - \frac{1}{x + 5}$
5) $\frac{7}{x^2 - 16} - \frac{3}{x - 4}$
The content is a single centered vertical list of five subtraction problems in the center of the page.
User: butterfly method
1. **Problem:** Subtract $\frac{5}{8}a - \frac{2}{8}a$.
Since both terms have the same denominator and variable, subtract the numerators:
$$\frac{5}{8}a - \frac{2}{8}a = \frac{5 - 2}{8}a = \frac{3}{8}a$$
2. **Problem:** Subtract $\frac{7}{3}x - \frac{2}{4}y$.
These terms have different variables and denominators, so they cannot be combined directly.
The butterfly method helps find a common denominator:
Multiply numerator of first fraction by denominator of second and vice versa:
$$\frac{7}{3}x - \frac{2}{4}y = \frac{7 \times 4}{3 \times 4}x - \frac{2 \times 3}{4 \times 3}y = \frac{28}{12}x - \frac{6}{12}y$$
Now subtract:
$$\frac{28}{12}x - \frac{6}{12}y = \frac{28x - 6y}{12}$$
3. **Problem:** Subtract $\frac{6}{x + 4} - \frac{2}{x - 1}$.
Use butterfly method to find common denominator:
Multiply numerator of first fraction by denominator of second and vice versa:
$$\frac{6}{x + 4} - \frac{2}{x - 1} = \frac{6(x - 1)}{(x + 4)(x - 1)} - \frac{2(x + 4)}{(x - 1)(x + 4)}$$
Subtract numerators:
$$\frac{6(x - 1) - 2(x + 4)}{(x + 4)(x - 1)} = \frac{6x - 6 - 2x - 8}{(x + 4)(x - 1)} = \frac{4x - 14}{(x + 4)(x - 1)}$$
4. **Problem:** Subtract $\frac{4}{x - 2} - \frac{1}{x + 5}$.
Use butterfly method:
$$\frac{4}{x - 2} - \frac{1}{x + 5} = \frac{4(x + 5)}{(x - 2)(x + 5)} - \frac{1(x - 2)}{(x + 5)(x - 2)}$$
Subtract numerators:
$$\frac{4x + 20 - (x - 2)}{(x - 2)(x + 5)} = \frac{4x + 20 - x + 2}{(x - 2)(x + 5)} = \frac{3x + 22}{(x - 2)(x + 5)}$$
5. **Problem:** Subtract $\frac{7}{x^2 - 16} - \frac{3}{x - 4}$.
Note that $x^2 - 16 = (x - 4)(x + 4)$.
Use butterfly method:
$$\frac{7}{(x - 4)(x + 4)} - \frac{3}{x - 4} = \frac{7}{(x - 4)(x + 4)} - \frac{3(x + 4)}{(x - 4)(x + 4)}$$
Subtract numerators:
$$\frac{7 - 3(x + 4)}{(x - 4)(x + 4)} = \frac{7 - 3x - 12}{(x - 4)(x + 4)} = \frac{-3x - 5}{(x - 4)(x + 4)}$$
**Final answers:**
1) $\frac{3}{8}a$
2) $\frac{28x - 6y}{12}$
3) $\frac{4x - 14}{(x + 4)(x - 1)}$
4) $\frac{3x + 22}{(x - 2)(x + 5)}$
5) $\frac{-3x - 5}{(x - 4)(x + 4)}$