Let's solve each of the given equations step-by-step.

Problem 1

Equation: $3x = 81 - 2x + 5$

Solution:

1. Combine like terms: $3x + 2x = 81 + 5$
2. Simplify: $5x = 86$
3. Divide by 5: $x = \frac{86}{5} = 17.2$

Problem 2

Equation: $2x + 1 = 32 - 2x + 5$

Solution:

1. Combine like terms: $2x + 2x = 32 + 5 - 1$
2. Simplify: $4x = 36$
3. Divide by 4: $x = 9$

Problem 3

Equation: $\sqrt{16(2x - 1)} = 64$

Solution:

1. Square both sides to remove the square root: $16(2x - 1) = 64^2$
2. Calculate $64^2 = 4096$: $16(2x - 1) = 4096$
3. Divide by 16: $2x - 1 = 256$
4. Solve for $x$: $2x = 257 \Rightarrow x = 128.5$

Problem 4

Equation: $25(3x + 1) = 125x + 3$

Solution:

1. Expand the left side: $75x + 25 = 125x + 3$
2. Move terms involving $x$ to one side and constants to the other: $75x - 125x = 3 - 25$
3. Simplify: $-50x = -22$
4. Divide by -50: $x = \frac{22}{50} = 0.44$

Problem 5

Equation: $7x + 5 = 1$

Solution:

1. Subtract 5 from both sides: $7x = -4$
2. Divide by 7: $x = -\frac{4}{7}$

Let me know if you'd like further details or explanations on any step.

Related Questions:

1. How do you solve equations involving square roots?
2. What are some methods for combining like terms effectively?
3. How do you isolate a variable in multi-step equations?
4. What is the significance of balancing equations in algebra?
5. How can you check if your solution is correct for an equation?

Tip: When solving equations, always perform the same operation on both sides to maintain equality.