Section 3: Equations of Lines

Elementary and Intermediate Algebra: Graphs & Models (3rd Edition)

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Section 3.3 Equations of Lines 271 Version: Fall 2007 3.3 Equations of Lines In this section we will develop the slope-intercept form of a line. When you have completed the work in this section, you should be able to look at the graph of a line and determine its equation in slope-intercept form. The Slope-Intercept Form In the previous section, we developed the formula for the slope of a line. Let’s assume that the dependent variable is y and the independent variable is x and we have a line passing through the points P ( x 1 , y 1 ) and Q ( x 2 , y 2 ) , as shown in Figure 1 . x y P ( x 1 ,y 1 ) Q ( x 2 ,y 2 ) Δ x = x 2 - x 1 Δ y = y 2 - y 1 Figure 1. Determining the slope of a line through two points. As we sweep our eyes from left to right, note that the change in x is x = x 2 x 1 and the change in y is y = y 2 y 1 . Thus, the slope of the line is determined by the formula Slope = y x = y 2 y 1 x 2 x 1 . (1) Now consider the line in Figure 2 . Suppose that we are given two facts about this line: 1. The point where the line crosses the y -axis (the y -intercept) is (0 , b ) . 2. The “slope” of the line is some number m . To find the equation of the line pictured in Figure 2 , select an arbitrary point Q ( x, y ) on the line, then compute the slope of the line using ( x 1 , y 1 ) = P (0 , b ) and ( x 2 , y 2 ) = Q ( x, y ) in the slope formula ( 1 ). Slope = y 2 y 1 x 2 x 1 = y b x 0 Copyrighted material. See: http://msenux.redwoods.edu/IntAlgText/ 1
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272 Chapter 3 Linear Functions Version: Fall 2007 x y P (0 ,b ) Q ( x,y ) Slope = m Figure 2. Find the equation of the line in slope-intercept form. Simplify. Slope = y b x We’re given that the slope is the number m , so substitute this number for the word “Slope” in the last result. m = y b x Multiply both sides of the last equation by x . mx = y b Add b to both sides of the last equation to obtain mx + b = y, or upon exchanging sides of the equation, y = mx + b. The above discussion leads to the following result. The Slope-Intercept Form of a Line . If the line L intercepts the y -axis at the point (0 , b ) and has slope m , then the equation of the line is y = mx + b. (2) This form of the equation of a line is called the slope-intercept form . The function defined by the equation f ( x ) = mx + b is called a linear function .
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Section 3.3 Equations of Lines 273 Version: Fall 2007 It is important to note two key facts about the slope-intercept form y = mx + b . The coefficient of x (the m in y = mx + b ) is the slope of the line. The constant term (the b in y = mx + b ) is the y -coordinate of the y -intercept (0 , b ) . Procedure for Using the Slope-Intercept Form of a Line . When given the slope of a line and the y -intercept of the line, use the slope-intercept form as follows: 1. Substitute the given slope for m in the formula y = mx + b . 2. Substitute the y -coordinate of the y -intercept for b in the formula y = mx + b .
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