Verified Answer :

  • Recognize the point-slope form equation: \(y - y_1 = m(x - x_1)\), where \(m\) is the slope.
  • Compare the given equation \((y-2) = 3(x-1)\) to the standard form.
  • Identify the coefficient \(m = 3\) as the slope of the line.
  • The final answer is \(\boxed{3}\).

Explanation

  1. Understanding the Point-Slope Form

The equation of a line provided is \((y-2) = 3(x-1)\). To determine the slope of this line, we recall the point-slope form of a linear equation, which is expressed as:
\[y - y_1 = m(x - x_1)\]
In this formula, \(m\) represents the slope of the line, and \((x_1, y_1)\) is a specific point on the line.

  1. Identifying the Slope

By comparing the given equation \((y-2) = 3(x-1)\) with the standard point-slope form \(y - y_1 = m(x - x_1)\), we can identify the corresponding components:

  • The value \(y_1\) is \(2\).
  • The value \(x_1\) is \(1\).
  • The value \(m\), which represents the slope, is the coefficient of the term \((x - x_1)\).

Looking at the equation, we can see that the coefficient of \((x - 1)\) is \(3\). Therefore, the slope of the line is \(3\).

  1. Conclusion

Since the slope \(m = 3\), the line represented by the equation \((y-2) = 3(x-1)\) has a slope of \(3\). This matches the fourth option provided in the problem. The final answer is \(\boxed{3}\).

Examples

Understanding the slope of a line is fundamental in various real-world scenarios. For example, if you are designing a wheelchair ramp, the slope is a crucial safety feature defined by the ratio of the vertical rise to the horizontal run. By identifying the slope from an equation, engineers and architects can ensure that structures comply with safety standards like the Americans with Disabilities Act (ADA), which often requires a slope of 1:12 (or approximately 0.083). Mastering this concept allows you to model real-world inclines and rates of change accurately.

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