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One of the most useful ways to write the equation of a line is the point-slope form. This form is especially helpful when you already know a point through which the line passes and the slope. The point-slope form of a line is represented by the equation \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) describes a point on the line and \(m\) represents the slope.

In the given exercise, the point is \((-2,3)\) and the slope \(m\) is \(-1\). When you substitute these values into the point-slope formula, it becomes \(y - 3 = -1(x + 2)\). By solving this equation further, you can make it easier to graph, especially by converting into a more familiar form like the slope-intercept form.

Understanding the point-slope form is crucial for effectively writing linear equations from given data and is a foundational tool in algebra.

After formulating the line's equation using the point-slope method, transforming it into the slope-intercept form \(y = mx + b\) can simplify graphing. This form highlights two critical components of a line: the slope \(m\) and the y-intercept \(b\).

In our example, after simplification, the equation \(y - 3 = -1(x + 2)\) converts to \(y = -x + 1\). Here, \(m = -1\) is the slope, indicating the line decreases 1 unit vertically for every 1 unit it moves horizontally, explaining a downward incline. The y-intercept \(b = 1\) signifies the point \((0,1)\) where the line crosses the y-axis.

The ease of use and straightforward interpretation of data provided by the slope-intercept form make it a favorite among students and educators for quick graphing and analysis of linear equations.

Once the equation is in slope-intercept form, graphing the line becomes more straightforward. This form allows you to quickly identify the starting point on a graph, the y-intercept, and then use the slope to determine other points on the line. Here's a step-by-step approach:

• Locate the y-intercept \((0, 1)\) on the graph. This is where the line will cross the y-axis and is a good starting point for drawing the line.
• Use the slope to find another point. With a slope of \(-1\), move down 1 unit and to the right 1 unit. Plot this second point at \((1, 0)\).
• Draw a line through the points \((0, 1)\) and \((1, 0)\). Extend the line in both directions to ensure it covers the graph space adequately.
• To verify, check that this line passes through the initial point \((-2, 3)\), confirming the line is accurately drawn.

Using these graphing techniques simplifies drawing a line while providing visual reinforcement of the relationships represented by a linear equation.