To find the answers to the given equations, let's solve each question systematically and provide an explanation:

7. $\frac{1}{2}x + 3 - \frac{1}{4}x = 3 + \frac{1}{4}x$

1. Combine like terms:

   • $\frac{1}{2}x - \frac{1}{4}x = \frac{2}{4}x - \frac{1}{4}x = \frac{1}{4}x$.
   • The equation becomes $\frac{1}{4}x + 3 = 3 + \frac{1}{4}x$.

2. Subtract $\frac{1}{4}x$ from both sides:

   • $3 = 3$.
     This shows the equation is true for all values of $x$.

Conclusion: The equation has infinitely many solutions.

8. $4 + 3x = 3x - 7$

1. Subtract $3x$ from both sides:

   • $4 = -7$.

2. Since $4 \neq -7$, there is no solution.

Conclusion: The equation is inconsistent, and there is no solution.

9. $5.4x + 12 = 2(2.7x - 9)$

1. Expand the right side:

   • $2(2.7x - 9) = 5.4x - 18$.
   • The equation becomes $5.4x + 12 = 5.4x - 18$.

2. Subtract $5.4x$ from both sides:

   • $12 = -18$.

3. Since $12 \neq -18$, there is no solution.

Conclusion: The equation is inconsistent, and there is no solution.

10. $6 + 2.5x = 0.5x - 8$

1. Subtract $0.5x$ from both sides:

   • $6 + 2.5x - 0.5x = -8$.
   • $6 + 2x = -8$.

2. Subtract 6 from both sides:

   • $2x = -14$.

3. Divide by 2:

   • $x = -7$.

Conclusion: The solution is $x = -7$.

11. $2(x - 1) + 6x = 4(2x - 1) + 2$

1. Expand both sides:

   • Left: $2x - 2 + 6x = 8x - 2 + 2$.
   • Right: $8x$.

2. Combine like terms:

   • $8x - 2 = 8x$.

3. Subtract $8x$ from both sides:

   • $-2 = 0$.

4. Since $-2 \neq 0$, there is no solution.

Conclusion: The equation is inconsistent, and there is no solution.

Summary:

• 7: Infinitely many solutions.
• 8: No solution.
• 9: No solution.
• 10: $x = -7$.
• 11: No solution.

Would you like me to expand on any part of this solution? Let me know if you have questions!

Relative Questions:

1. What does it mean for an equation to have "infinitely many solutions"?
2. How can we identify when an equation has no solution?
3. What is the importance of simplifying terms first in solving equations?
4. Why is it necessary to check your solution after solving?
5. Can you show a real-life scenario where such equations might arise?

Tip:

Always simplify terms on both sides of the equation before proceeding to solve—this minimizes errors and makes calculations clearer.