The equation given is:

$5(2x + 3) - 7x = 21$

Let's solve it step by step:

Step 1: Expand the expression

Distribute the $5$ across the terms inside the parentheses:

$5(2x + 3) = 10x + 15$

So the equation becomes:

$10x + 15 - 7x = 21$

Step 2: Combine like terms

Now, combine the $x$-terms:

$(10x - 7x) + 15 = 21$

This simplifies to:

$3x + 15 = 21$

Step 3: Isolate the variable

Subtract 15 from both sides:

$3x = 21 - 15$

$3x = 6$

Step 4: Solve for $x$

Finally, divide both sides by 3:

$x = \frac{6}{3} = 2$

Thus, the solution is:

$x = 2$

Let me know if you need further clarification or details on any of the steps.

Here are some related questions to explore further:

1. What would happen if the equation had a different constant term on the right-hand side?
2. How would you solve the equation if there were fractions involved instead of integers?
3. Can you solve for $x$ if there was no solution (e.g., parallel lines in a system of equations)?
4. What if the equation was quadratic (e.g., had $x^2$ terms)? How would that change the solution process?
5. How do you verify the solution by substituting back into the original equation?

Tip: Always check your final answer by plugging it back into the original equation to ensure it's correct.