Let's solve the equation step by step:

\[
-8(t - 4) + 7 = 4[t - 3(1 - t)] + 6
\]

### Step 1: Distribute on both sides

On the left-hand side:
\[
-8(t - 4) = -8t + 32
\]
So, the equation becomes:
\[
-8t + 32 + 7 = 4[t - 3(1 - t)] + 6
\]
\[
-8t + 39 = 4[t - 3(1 - t)] + 6
\]

On the right-hand side, distribute within the brackets:
\[
t - 3(1 - t) = t - 3 + 3t = 4t - 3
\]
So, the equation becomes:
\[
-8t + 39 = 4(4t - 3) + 6
\]
Now, distribute the 4:
\[
4(4t - 3) = 16t - 12
\]
Thus, the equation becomes:
\[
-8t + 39 = 16t - 12 + 6
\]
Simplify the right-hand side:
\[
-8t + 39 = 16t - 6
\]

### Step 2: Combine like terms

Move all terms involving \(t\) to one side and constant terms to the other side:
\[
-8t - 16t = -6 - 39
\]
Simplify:
\[
-24t = -45
\]

### Step 3: Solve for \(t\)

Divide both sides by \(-24\):
\[
t = \frac{-45}{-24} = \frac{45}{24}
\]

Simplify the fraction:
\[
t = \frac{15}{8}
\]

Thus, the solution is:
\[
t = \frac{15}{8}
\]

Would you like further details or have any questions? Here are 5 related questions:

1. How would you check if \(t = \frac{15}{8}\) is correct by plugging it back into the equation?
2. What is the process of distributing numbers over brackets?
3. How do you handle negative coefficients when solving equations?
4. Can you solve a similar equation where fractions are involved from the start?
5. How would the equation change if there were additional terms involving \(t^2\)?

**Tip:** Always simplify both sides of an equation as much as possible before moving terms around.

This detailed solution guides you through solving the linear equation -8(t-4)+7=4[t-3(1-t)]+6 step-by-step. Learn key algebraic concepts, including the distributive property, combining like terms, and simplifying fractions. Ideal for students in Grades 8-10.

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