Let's solve the equation step by step.

We are given:

\[
-2(8x + 1) - 1 = -7(2x - 3) - x
\]

### Step 1: Distribute the constants on both sides of the equation.

\[
-2(8x + 1) = -16x - 2
\]
So, the left-hand side becomes:

\[
-16x - 2 - 1 = -16x - 3
\]

Now, for the right-hand side:

\[
-7(2x - 3) = -14x + 21
\]
So the right-hand side becomes:

\[
-14x + 21 - x = -15x + 21
\]

Now the equation looks like:

\[
-16x - 3 = -15x + 21
\]

### Step 2: Combine like terms and solve for \(x\).

Move all terms involving \(x\) to one side of the equation and the constants to the other side. Start by adding \(16x\) to both sides:

\[
-3 = -15x + 16x + 21
\]

This simplifies to:

\[
-3 = x + 21
\]

Now, subtract 21 from both sides:

\[
-3 - 21 = x
\]

\[
x = -24
\]

### Final Answer:
\[
x = -24
\]

Would you like further details or have any questions?

Here are 5 related questions to explore further:
1. How would you check if \(x = -24\) is the correct solution?
2. What happens when you distribute negative numbers across parentheses in equations?
3. How would this equation change if the coefficient of \(x\) in the first term was different?
4. Can you solve similar equations that include fractions?
5. How do you graph the solution of an equation like this?

**Tip:** Always check your solution by plugging the value of \(x\) back into the original equation.

−2(8x + 1) − 1 = −7(2x − 3) − x

This detailed solution walks through solving the linear equation −2(8x + 1) − 1 = −7(2x − 3) − x step by step. Learn about using the distributive property and combining like terms. Perfect for students in Grades 8-10.

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