Substitution in Systems of Nonlinear Equations — Practice problem

Justin Spence finally bought his own place. It wasn't as big or as lavish as he had hoped but it was his. The house was isolated on top of a hill near a small suburb. It had six previous owners in the course of three years, but it was cheap, and Justin could find nothing wrong with the property.

That night, before going to bed, his house began to shake. He could hear a low rumble coming from below him. "An earthquake!" he exclaimed, running for the nearest door frame. But it stopped as quickly as it had started.

The next day he turned on the news as he ate breakfast. "And in other news, Expii County was struck with a small earthquake last night around midnight. This is its first documented earthquake within the past decade. Experts are unsure what caused it."

That day, at work, Justin was talking to his friend Sam in the break room. "How crazy was that earthquake last night?" Justin said.

"Earthquake?" Sam asked.

"You didn't feel it? It was on the news this morning."

Sam took out her phone. After a couple seconds of typing she looked back up at Justin. "There was no earthquake last night Justin."

"What are you talking about? I felt it. I watched them talk about it on the news!"

"I just looked it up and the last documented earthquake hear was over five years ago."

"Maybe it hasn't updated yet," Justin offered up.

That night, around midnight, his house began to shake again. This time it was more intense. His dishes began to violently rattle in the cabinets and a few books toppled off his bookcase. He ran to the nearest door frame. But just like before, it ended as suddenly as it began.

The next morning he turned on the news. "Last night Expii County experienced yet another earthquake. This one measuring a $4.0$ on the Richter Scale. Experts are still unsure why these are happening."

When Justin stepped outside he felt that something was off immediately. When he arrived at work he found the building abandoned. He quickly called Sam. "Sam, why is our building abandoned?"

"Justin!? Where have you been?"

"What are you talking about?" His unease began to creep deeper into his stomach.

"You've been missing for four years! Where have you been? We gave up looking for you awhile ago. Most of us assumed the worse."

"Four years? Sam, are you messing with me? I saw you yesterday. We talked about that earthquake."

"That's the last thing you remember?"

"Yes. Then I went home and there was another earthquake. I woke up and called you."

"Well we definitely didn't have an earthquake this morning. In fact, no earthquakes the last four years either."

"No earthquakes? It was registered as a $4.0$ on the Richter scale!"

"I don't know what to tell you Justin. You can't just show up after being missing for $4$ years and expect to pretend like nothing happened. Look, I'm glad you're alive, but I need to go. Maybe we can meet up tomorrow."

That night his house began to shake again, more intensely than before. Bookcases toppled over. His dishes sprung from the cabinets and shattered on the ground. A loud groaning erupted from beneath the ground. And then it was over.

The next morning Justin hesitantly turned on the TV. "For the third night in a row Expii County experienced an earthquake but this the most devastating. We're getting reports that its $403$ times more intense than yesterday's. Still no word on what caused this or what this holds for the future."