When working with linear equations, particularly in point-slope form, it's essential to understand how to derive the slope when only two points are provided. The point-slope form of a line is expressed as:

\(y - y_1 = m(x - x_1)\)

In this equation, \(m\) represents the slope, while \((x_1, y_1)\) is a point on the line. If the slope \(m\) is not given, it can be calculated using the coordinates of the two points. For example, if the points are \((-1, -5)\) and \((2, 4)\), the slope can be determined using the formula:

\(m = \frac{y_2 - y_1}{x_2 - x_1}\)

Substituting the values from the points, we have:

\(m = \frac{4 - (-5)}{2 - (-1)} = \frac{4 + 5}{2 + 1} = \frac{9}{3} = 3\)

Now that we have the slope \(m = 3\), we can substitute this value back into the point-slope form. Choosing one of the points, say \((-1, -5)\), we can write:

\(y - (-5) = 3(x - (-1))\)

This simplifies to:

\(y + 5 = 3(x + 1)\)

This equation represents the line that passes through the two given points. To graph this line, you can plot the points \((-1, -5)\) and \((2, 4)\) on a coordinate plane and connect them with a straight line. The key takeaway is that regardless of which point you choose as \((x_1, y_1)\), the resulting equation will represent the same line.