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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. In this chapter the student is shown how graphs provide information that is not always evident from the equation alone. The chapter begins by establishing the relationship between the variables in an equation, the number of coordinate axes necessary to construct its graph, and the spatial dimension of both the coordinate system and the graph. Interpretation of graphs is also emphasized throughout the chapter, beginning with the plotting of points. The slope formula is fully developed, progressing from verbal phrases to mathematical expressions. The expressions are then formed into an equation by explicitly stating that a ratio is a comparison of two quantities of the same type (e.g., distance, weight, or money). This approach benefits students who take future courses that use graphs to display information.The student is shown how to graph lines using the intercept method, the table method, and the slope-intercept method, as well as how to distinguish, by inspection, oblique and horizontal/vertical lines. Objectives of this module: be able to find the equation of a line using either the slope-intercept form or the point-slope form of a line.

Overview

  • The Slope-Intercept and Point-Slope Forms

The slope-intercept and point-slope forms

In the pervious sections we have been given an equation and have constructed the line to which it corresponds. Now, however, suppose we're given some geometric information about the line and we wish to construct the corresponding equation. We wish to find the equation of a line.

We know that the formula for the slope of a line is m = y 2 - y 1 x 2 - x 1 . We can find the equation of a line using the slope formula in either of two ways:

If we’re given the slope, m , and any point ( x , 1 y 1 ) on the line, we can substitute this information into the formula for slope.
Let ( x , 1 y 1 ) be the known point on the line and let ( x , y ) be any other point on the line. Then

m = y - y 1 x - x 1 Multiply  both  sides  by x - x 1 . m ( x - x 1 ) = ( x - x 1 ) · y - y 1 x - x 1 m ( x - x 1 ) = y - y 1 For  convenience,  we'll  rewrite  the  equation . y - y 1 = m ( x - x 1 )

Since this equation was derived using a point and the slope of a line, it is called the point-slope form of a line.

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If we are given the slope, m , y-intercept ,  ( 0 , b ) , we can substitute this information into the formula for slope.
Let ( 0 , b ) be the y-intercept  and  ( x , y ) be any other point on the line. Then,

m = y - b x - 0 m = y - b x Multiply  both  sides  by  x . m · x = x · y - b x m x = y - b Solve  for  y . m x + b = y For  convenience,   we'll  rewrite  this  equation . y = m x + b

Since this equation was derived using the slope and the intercept, it was called the slope-intercept form of a line.

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We summarize these two derivations as follows.

Forms of the equation of a line

We can find the equation of a line if we’re given either of the following sets of information:
  1. The slope, m , and the y -intercept,  ( 0 , b ) , by substituting these values into

    y = m x + b

    This is the slope-intercept form.

  2. The slope, m , and any point, ( x 1 , y 1 ) , by substituting these values into

    y - y 1 = m ( x - x 1 )

    This is the point-slope form.

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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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