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The original price of a motorcycle in a straight-line depreciation model is a crucial part. Straight-line depreciation helps us understand how a motorcycle's value decreases over time. In the given equation for depreciation, \( y = -2150x + 17200 \), the original price is identified by focusing on the intercept, which is the code word for a number that's not tied to any variable.
The intercept, \( 17200 \), represents the motorcycle's value when it hits the road for the first time. This is essentially the price tag the motorcycle had before any wear and tear began to take its toll.

• The original price is found at \( x = 0 \).
• This means when the motorcycle hasn't experienced any years of depreciation, it's at its full original price.
• So, simply put, the original price is exactly \( 17200 \) dollars.

Understanding this helps to appreciate the true starting value of a motorcycle before depreciation starts eating into its worth.

The value loss each year in straight-line depreciation is represented by the coefficient that pairs with the variable \( x \) within the formula. In our equation, \( y = -2150x + 17200 \), we see \( -2150 \) right next to \( x \).
This is no ordinary number; it's a guide showing us exactly how much value the motorcycle is losing as each year ticks by.

• The \( -2150 \) tells us that annually, the motorcycle's value decreases by \( 2150 \) dollars.
• The negative sign simply means each year it's losing that amount, not gaining.

Two significant takeaways are:

• The constant loss of \( 2150 \) means we can expect a steady and predictable reduction in value for every year.
• Knowing this helps in planning for future resale or determining the worth of the motorcycle as time progresses.

Essentially, it's a steady climb downhill in terms of value, mapped out neatly each year.

Equation solving in straight-line depreciation allows us to pinpoint when a motorcycle's value hits rock bottom. Specifically, when its value becomes zero. The equation \( y = -2150x + 17200 \) gets us there by setting \( y = 0 \).
Steps to solve include:

• Set up the equation: \( 0 = -2150x + 17200 \).
• The goal is to solve for \( x \), which tells us the number of years for the motorcycle to fully depreciate.
• Rearrange the equation to isolate \( x \): \( 2150x = 17200 \).
• This transforms into: \( x = \frac{17200}{2150} \).
• Upon division, we find \( x = 8 \).

Therefore, the motorcycle will depreciate completely in 8 years. Knowing this provides clarity on the timeline of the motorcycle's value depleting entirely, which is crucial for budgeting and investment timelines.