To determine whether the equation \( \frac{1}{3}(t + 6) - 10 = -3t + 2 \) has one solution, we need to solve the equation for \( t \).

### Step 1: Distribute the \( \frac{1}{3} \) across the terms inside the parentheses.
\[
\frac{1}{3}(t + 6) = \frac{1}{3}t + \frac{6}{3} = \frac{1}{3}t + 2
\]
So the equation becomes:
\[
\frac{1}{3}t + 2 - 10 = -3t + 2
\]

### Step 2: Simplify the left side of the equation.
\[
\frac{1}{3}t - 8 = -3t + 2
\]

### Step 3: Move all terms involving \( t \) to one side and constant terms to the other side.
\[
\frac{1}{3}t + 3t = 2 + 8
\]
\[
\frac{1}{3}t + 3t = 10
\]

### Step 4: Combine like terms.
To combine \( \frac{1}{3}t \) and \( 3t \), note that \( 3t = \frac{9}{3}t \):
\[
\frac{1}{3}t + \frac{9}{3}t = \frac{10}{3}t
\]
So the equation is now:
\[
\frac{10}{3}t = 10
\]

### Step 5: Solve for \( t \).
Multiply both sides by the reciprocal of \( \frac{10}{3} \) which is \( \frac{3}{10} \):
\[
t = 10 \times \frac{3}{10} = 3
\]

### Conclusion:
The solution is \( t = 3 \). Since we found a specific value for \( t \), the equation has **one solution**.

Would you like more details or have any questions?

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Here are some related questions:
1. What does it mean for an equation to have no solution?
2. How can you check if an equation has infinitely many solutions?
3. How do you solve a quadratic equation?
4. What happens when you move all terms to one side of an equation?
5. How do you simplify an equation involving fractions?

**Tip:** Always check your solution by substituting it back into the original equation.

Learn how to solve the equation (1/3)(t + 6) - 10 = -3t + 2 step-by-step with detailed solutions. This problem covers key concepts in solving linear equations. Suitable for students in Grades 7-9.

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