Let's go through Liam's steps to find the mistake. The original equation is:
\[ 7x + 6(x + 3) = 83. \]
- The first step is correct: \[ 7x + 6(x + 3) = 7x + 6x + 18 = 83 \quad \text{(Liam should write 18 instead of 3)} \] So, he should have: \[ 7x + 6x + 18 = 83. \]
- Combining the \(7x\) and \(6x\): \[ 13x + 18 = 83. \]
- Subtracting 18 from both sides: \[ 13x = 83 - 18. \] Calculating \(83 - 18\) gives \(65\), not \(80\). So, the correct equation here would be: \[ 13x = 65. \]
- Dividing both sides by 13: \[ x = \frac{65}{13} = 5. \]
Liam's mistake occurred in these steps where he miscalculated \(83 - 18\) as \(80\) instead of \(65\). Thus, the correct solution for \(x\) is \(5\).