Question: User: 2e-3-\frac{1}{5}e =\frac{7}{5}+3 e +4
1. **State the problem:** Solve the equation $$2e - 3 - \frac{1}{5}e = \frac{7}{5} + 3e + 4$$ for the variable $e$.
2. **Write down the equation:**
$$2e - 3 - \frac{1}{5}e = \frac{7}{5} + 3e + 4$$
3. **Combine like terms on the left side:**
$$2e - \frac{1}{5}e = \frac{10}{5}e - \frac{1}{5}e = \frac{9}{5}e$$
So the left side becomes:
$$\frac{9}{5}e - 3$$
4. **Combine constants on the right side:**
$$\frac{7}{5} + 4 = \frac{7}{5} + \frac{20}{5} = \frac{27}{5}$$
So the right side becomes:
$$\frac{27}{5} + 3e$$
5. **Rewrite the equation:**
$$\frac{9}{5}e - 3 = \frac{27}{5} + 3e$$
6. **Bring all terms involving $e$ to one side and constants to the other:**
$$\frac{9}{5}e - 3e = \frac{27}{5} + 3$$
7. **Simplify the left side:**
$$\frac{9}{5}e - \frac{15}{5}e = -\frac{6}{5}e$$
8. **Simplify the right side:**
$$\frac{27}{5} + 3 = \frac{27}{5} + \frac{15}{5} = \frac{42}{5}$$
9. **Rewrite the equation:**
$$-\frac{6}{5}e = \frac{42}{5}$$
10. **Solve for $e$ by dividing both sides by $-\frac{6}{5}$:**
$$e = \frac{\frac{42}{5}}{-\frac{6}{5}} = \frac{42}{5} \times \frac{5}{-6} = \frac{42}{\cancel{5}} \times \frac{\cancel{5}}{-6} = \frac{42}{-6} = -7$$
**Final answer:**
$$e = -7$$