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Calculating the slope of a line is an essential step in determining the point-slope form. The slope, often referred to by the letter \(m\), represents the steepness or incline of a line. It is calculated by looking at two points through which the line passes. Given points \((-2, 3)\) and \(1, 0)\), the slope is computed using the formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
This means subtracting the y-value of the first point from the y-value of the second point, and similarly for the x-values.
So, for the points provided, it looks like this:

• \(y_2 - y_1 = 0 - 3 = -3\)
• \(x_2 - x_1 = 1 - (-2) = 1 + 2 = 3\)

Now, dividing these values gives \(m = \frac{-3}{3} = -1\).
This indicates a line with a downward slope. When moving from left to right, the line decreases one unit for each unit moved horizontally.

Linear equations form the core structure for describing straight lines through algebraic expressions. The point-slope form of a linear equation is particularly helpful when a specific point on the line and the slope are known. The general formula for point-slope form is: \(y - y_1 = m(x - x_1)\).
By substituting the slope \(m\) and a specific point \((x_1, y_1)\) from the problem, this form makes it straightforward to derive the complete linear equation of the line. In our exercise, using the point \((-2, 3)\) and slope \(-1\):

• Substitute \(y_1 = 3\) and \(x_1 = -2\)
• Substitute \(m = -1\)

This substitution gives the equation: \(y - 3 = -1(x + 2)\).
This point-slope equation captures all the necessary information about the linear path, making it a practical representation when graphing or analyzing the line further.

Graphing the line derived from a point-slope equation visually illustrates the relationship between the points and slope calculated. Start by plotting the given points \((-2, 3)\) and \(1, 0)\) on a graph.
Make sure you label both axes accurately so that points are plotted correctly. Once the points are on the graph, the next step is to draw the line connecting these points, illustrating the equation \(y - 3 = -1(x + 2)\).
This line should reflect the calculated slope, indicating a decline at a rate of -1, meaning for every block moved horizontally, the line will move down one unit vertically.
Using the graph allows you to see how each mathematical component contributes to the overall linear equation's graphical representation. This kind of visualization aids in understanding not just how the slope works, but also how lines behave in a coordinate plane.