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The slope of a line is a measure of its steepness and direction. It tells us how much y changes for a change in x. When you have two points on a line, such as

• (6, 5)
• (7, 1)

you can find the slope using the formula: \\( m = \frac{y_2 - y_1}{x_2 - x_1} \).

For this pair of points, the slope \( m \) is calculated as follows:

• Subtract the y-coordinates: \(1 - 5 = -4\)
• Subtract the x-coordinates: \(7 - 6 = 1\)

So, \\( m = \frac{-4}{1} = -4 \).

A negative slope indicates that the line is going downwards as it moves from left to right. In our case, the slope \( -4 \) tells us that for each unit increase in x, y decreases by 4 units. Understanding slope is crucial as it lays the foundation for writing equations of lines in various forms.

Once we have the slope of a line, the next step is to write the equation of the line. There are different forms for the equation of a line, but one of the most useful is the **point-slope form**. The formula is:\( y - y_1 = m(x - x_1) \)

Here, \( m \) represents the slope, and \((x_1, y_1)\) is a point on the line. For our exercise, \( m = -4 \) and we can use point (6,5). Placing these into the formula gives us:

• \( y - 5 = -4(x - 6) \)

This is the point-slope form equation of the line passing through the points (6,5) and (7,1).

This equation form is very powerful because it immediately gives you the slope and a point on the line, making it easier to graph or further manipulate into other forms like slope-intercept form.

Solving mathematical problems involves a step-by-step process where each step builds upon the previous one. In the context of writing equations of lines, consider the sequence:

• Identify given data
• Use formulas to calculate necessary values
• Substitute these values into relevant equations

For instance, when asked to find a line's equation through two points, you might first find the slope. Then, choose a formula, like point-slope, to write the equation.

This process helps in organizing your calculations and ensuring accuracy. It's crucial to understand each step fully, as mastering these fundamentals will enhance your problem-solving skills not just in geometry, but in all areas of mathematics. Emphasizing clarity and precision in each step aids in avoiding errors and understanding the concepts in depth.