Conversions: Point-Slope and Slope-Intercept Form
Conversions between point-slope form and slope-intercept form are essential concepts in understanding linear equations, which are fundamental in various disciplines including art, engineering, and mathematics. The point-slope form, expressed as \( y - y_1 = m(x - x_1) \), is particularly useful when the slope of a line and a specific point on that line are known. On the other hand, the slope-intercept form, given by \( y = mx + b \), directly reveals the slope and the y-intercept of the line, making it straightforward to graph.
To convert between the two forms, algebraic manipulation is employed. For example, starting from point-slope form allows one to derive the slope-intercept form by isolating \( y \). Conversely, if the slope and the y-intercept are known, one can easily write the equation in slope-intercept form. These conversions enhance the ability to interpret and utilize linear equations effectively, whether for theoretical explorations or practical applications in various fields. Understanding these forms and their interrelationships is fundamental for anyone engaging with the concept of linear relationships.
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Conversions: Point-Slope and Slope-Intercept Form
Point-slope form (y − y1 = m(x − x1) and slope-intercept form (y = mx + b) are two of the most common ways to describe a line. The two can be converted into one another through the use of algebraic principles.
One can scarcely open a book on art, engineering, architecture, or mathematics and not find a multitude of instances of one of the most important of geometric figures: the straight line. A line is the set of all points that are an equal distance from two fixed points and . See Figure 1.
The key word in our definition is distance. The distance between two points and is the given by the distance formula: . See Figure 2.
By applying the distance formula to the equal distances in Figure 1 and carefully working through several steps of algebra, the following equation is obtained.
Here and are numbers. Equation is the general form of the equation of a line. Two particular forms of the equation of a line are the the slope-intercept form (y = mx + b); and the point-slope form (y − y1 = m(x − x1), and it is possible to convert from the former to the latter.
Overview
If , then can be solved for and cast in the following form.
Here and are numbers. It can be shown that is the slope (rise over run) of the line, and that is the y-coordinate of the y-intercept of the line. See Figure 3.
Equation is called the slope-intercept form of the equation of a line, because both the slope and the y-intercept of the line can be easily read from the equation in this form. The usefulness of this fact is illustrated in the following two examples.
If the equation of a line is , it can be immediately seen that and . Therefore the graph of the equation is a line whose slope is 2 and whose y-intercept is the point .
If it is known that the slope of a line is and that its y-intercept is the point , it is readily seen that and . So it can immediately be concluded that the equation of the line is .
Suppose both the slope of a line and a point on the line that is not the y-intercept are known, leaving unknown. Another form of the equation of a line is called for, namely the point-slope form. This form is based on the notion that, provided the slope of a line is defined, it will have the same value when it is computed using a fixed point and any variable point , as long as . See Figure 4 below.
The equation of the line can be written by simply applying the definition of slope to the two points labeled in Figure 4.
Or, multiplying both sides of by one obtains:
Notice that unlike , is valid for all . As an example, suppose it is known that a line has slope and that it passes through the point . The equation can be written down immediately in point-slope form: .
When the equation of a line is in point-slope form , the y-intercept can be determined by manipulating the equation into form . This is done by distributing m over the binomial and then adding . When these steps are done for the equation in the last example, the equation is manipulated into , and it is clear that the y-intercept of the line is the point . See Figure 5.
Bibliography
Clark, David M. Euclidean Geometry: A Guided Inquiry Approach. Berkeley: Amer.Mathematical Soc., 2012. Print.
Ostermann, Alexander, and Gerhard Wanner. Geometry by Its History. New York: Springer, 2012. Print.