12−[3+4m−6(3m−2)]=−7(2m−8)−3[(m−2)+3m−5]

To solve the equation:

\[
12 - [3 + 4m - 6(3m - 2)] = -7(2m - 8) - 3[(m - 2) + 3m - 5]
\]

we’ll proceed by simplifying both sides step-by-step.

### Step 1: Expand the expressions inside the brackets
1. Expand \(6(3m - 2)\):
   \[
   6(3m - 2) = 18m - 12
   \]
   Substitute this into the equation:
   \[
   12 - [3 + 4m - (18m - 12)]
   \]

2. Simplify inside the brackets on the left side:
   \[
   3 + 4m - 18m + 12 = 15 - 14m
   \]
   So, the left side becomes:
   \[
   12 - (15 - 14m)
   \]
   Distribute the negative sign:
   \[
   12 - 15 + 14m = -3 + 14m
   \]

3. Expand \(-7(2m - 8)\) on the right side:
   \[
   -7(2m - 8) = -14m + 56
   \]

4. Expand \(3[(m - 2) + 3m - 5]\):
   \[
   (m - 2) + 3m - 5 = 4m - 7
   \]
   Then:
   \[
   3(4m - 7) = 12m - 21
   \]
   So the right side becomes:
   \[
   -14m + 56 - (12m - 21)
   \]
   Simplify:
   \[
   -14m + 56 - 12m + 21 = -26m + 77
   \]

### Step 2: Substitute back to get a simpler equation
After simplification, our equation is:
\[
-3 + 14m = -26m + 77
\]

### Step 3: Collect like terms
Add \(26m\) to both sides:
\[
-3 + 40m = 77
\]
Add 3 to both sides:
\[
40m = 80
\]
Divide by 40:
\[
m = 2
\]

### Final Answer:
\[
m = 2
\]

Would you like a further breakdown of any steps?

---

Here are some related questions to explore:
1. How would the solution change if we had a different constant on the right side?
2. How can we verify the solution by substituting back into the original equation?
3. What if the equation had fractions instead of integers?
4. Can we solve similar equations using matrix methods?
5. What are some practical applications of solving linear equations like this?

**Tip:** Always simplify complex expressions inside brackets first to make the equation easier to work with.

Follow a detailed solution to solve the complex linear equation 12−[3+4m−6(3m−2)]=−7(2m−8)−3[(m−2)+3m−5]. Learn key algebraic concepts like equation simplification and using the distributive property. Suitable for students in Grades 8-10.

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