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Extra info for Algebra II (Cliffs Quick Review)
Solve this equation for y. Ax + By = C By = –Ax + C y = - BA x + CB The value - AB becomes the slope of the line, and CB becomes the y-intercept value. If - AB is replaced with m and CB is replaced with b, the equation becomes y = mx + b. This is known as the slope-intercept form of a nonvertical line. Example 11: Find the slope and y-intercept value of the line with the equation 3x – 4y = 20. Solve for y. 3x – 4y = 20 –4y = –3x + 20 y = 34 x – 5 Therefore, the slope of the line is 34 , and the y-intercept value is –5.
This means that the “(0, 0) side” of the boundary line is the desired region to be shaded. Now, shade that region as shown in Figure 2-12. F 4/19/01 8:50 AM Page 37 Chapter 2: Segments, Lines, and Inequalities 37 Figure 2-12 The shading is below the line. y 4 3 3x + 4y <12 2 1 −3 −2 −1 1 2 3 x 3x + 4y = 12 4 −1 −2 −3 Example 14: Graph y ≥ 2x + 3. First, graph y = 2x + 3 (see Figure 2-13). Figure 2-13 This boundary is solid. y 5 (1,5) 4 3 (0,3) 2 (−1,1) −3 −2 −1 1 1 2 3 x −1 −2 y = 2x + 3 −3 Notice that the boundary is a solid line, because the original inequality is ≥.
Example 4: A line passes through (–5,8) with a slope of 23 . If another point on this line has coordinates (x,12), find x. y 2 - y1 m = x 2 - x1 2 = 12 - 8 3 x - ^ - 5h 2= 4 3 x+5 2 ^ x + 5h = 4 ^ 3 h 2x + 10 = 12 2x = 2 x=1 Slope of Parallel and Perpendicular Lines Parallel lines have equal slopes. Conversely, if two different lines have equal slopes, they are parallel. F 4/19/01 8:50 AM Page 29 Chapter 2: Segments, Lines, and Inequalities 29 their slopes are negative reciprocals (actually, opposite reciprocals) of one another, or the product of their slopes is –1.