Let's solve the equation step by step:

### Step 1: Expand both sides of the equation
Expand the terms on both sides:

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

On the left-hand side:

\[
8t + 8a - 3t + 12 = 6(t - 7) + 8
\]

On the right-hand side, expand \(6(t - 7)\):

\[
8t + 8a - 3t + 12 = 6t - 42 + 8
\]

### Step 2: Simplify the equation
Simplify both sides:

Left side:

\[
(8t - 3t) + 8a + 12 = 5t + 8a + 12
\]

Right side:

\[
6t - 42 + 8 = 6t - 34
\]

The equation becomes:

\[
5t + 8a + 12 = 6t - 34
\]

### Step 3: Move terms with \(t\) to one side
Subtract \(5t\) from both sides:

\[
8a + 12 = t - 34
\]

### Step 4: Solve for \(t\)
Now, isolate \(t\) by adding \(34\) to both sides:

\[
t = 8a + 12 + 34
\]

Simplify:

\[
t = 8a + 46
\]

So, the solution is:

\[
t = 8a + 46
\]

Would you like further explanation on any step?

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Here are 5 related questions for practice:

1. What would be the solution if \( a = 2 \)?
2. How would the equation change if \( -3(t - 4) \) became \( +3(t - 4) \)?
3. Can you solve \( 4(t + 3) - 2(t - 5) = 5(t - 6) + 10 \)?
4. How would you solve the equation if there was a different coefficient for \( a \)?
5. What is the value of \( t \) if \( a = -5 \)?

**Tip**: Always expand terms carefully to avoid sign errors when distributing.

Step-by-step solution to solve the linear equation 8(t + a) - 3(t - 4) = 6(t - 7) + 8 using the distributive property and variable isolation techniques. Learn how to simplify and solve for t. Suitable for students in Grades 8-10.

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