Let's solve the equation step by step:

### Given equation:
\[
4(3x + 2) - 5(6x - 1) = 2(x - 8) - 6(7x - 4)
\]

### Step 1: Expand both sides.
Distribute the constants across the parentheses:

\[
4(3x + 2) = 12x + 8
\]
\[
-5(6x - 1) = -30x + 5
\]
\[
2(x - 8) = 2x - 16
\]
\[
-6(7x - 4) = -42x + 24
\]

Now substitute these expanded terms back into the equation:

\[
(12x + 8) - (30x - 5) = (2x - 16) - (42x + 24)
\]

### Step 2: Simplify both sides.
Simplify the left-hand and right-hand sides:

\[
12x + 8 - 30x + 5 = 2x - 16 - 42x - 24
\]
\[
-18x + 13 = -40x - 40
\]

### Step 3: Combine like terms.
Now move all the terms involving \( x \) to one side, and constant terms to the other side:

Add \( 40x \) to both sides:

\[
-18x + 40x + 13 = -40
\]
\[
22x + 13 = -40
\]

Subtract \( 13 \) from both sides:

\[
22x = -40 - 13
\]
\[
22x = -53
\]

### Step 4: Solve for \( x \).
Now divide both sides by 22:

\[
x = \frac{-53}{22}
\]

Thus, the solution is:
\[
x = \frac{-53}{22}
\]

Would you like more details or have any questions?

Here are 5 questions related to this problem:
1. How would the equation change if the coefficients were negative?
2. What happens if we multiply the entire equation by a constant first?
3. Can this equation have multiple solutions?
4. How would the solution change if the equation were set equal to zero?
5. How can we verify the solution by substituting \( x = \frac{-53}{22} \) back into the original equation?

**Tip:** Always distribute carefully when solving linear equations, and make sure to combine like terms properly to avoid mistakes.

Learn how to solve the linear equation 4(3x + 2) - 5(6x - 1) = 2(x - 8) - 6(7x - 4) with a detailed step-by-step solution. This problem covers key algebraic concepts such as the distributive property and combining like terms. Suitable for students in Grades 8-10.

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