Let's solve the equation step by step:

Given:

$3(7x + 7) = 7(4x + 10)$

1. Distribute the 3 on the left-hand side: $3 \times 7x + 3 \times 7 = 21x + 21$

2. Distribute the 7 on the right-hand side: $7 \times 4x + 7 \times 10 = 28x + 70$

Now, the equation looks like this:

$21x + 21 = 28x + 70$

3. Move all the terms involving $x$ to one side: Subtract $21x$ from both sides:

$21 = 28x - 21x + 70$

Simplifying:

$21 = 7x + 70$

4. Move the constant terms to the other side: Subtract 70 from both sides:

$21 - 70 = 7x$

$-49 = 7x$

5. Solve for $x$: Divide both sides by 7:

$x = \frac{-49}{7} = -7$

So, the solution is:

$x = -7$

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Would you like a more detailed explanation or have any further questions? Here are 5 follow-up questions to help expand on the concept:

1. How would the solution change if the equation had been $5(7x + 3) = 4(3x + 6)$?
2. What happens if the equation has fractions, e.g., $\frac{1}{3}(7x + 2) = \frac{2}{5}(5x + 1)$?
3. How would you solve an equation with variables on both sides and a negative coefficient, like $-3(4x + 2) = 5(2x - 3)$?
4. What is the effect of multiplying both sides of an equation by a negative number?
5. How would you check the solution for $x = -7$ in the original equation?

Tip: Always distribute the constants carefully and combine like terms on both sides to simplify the equation before isolating the variable.