Unformatted text preview: he equation of a line based on the line’s graph. www.petersons.com 284 Chapter 16 Example: In the xyplane, what is the slope of the line defined by the two points P(2,1) and Q(–3,4)? (A) –3 (B) (C) (D) (E) Solution: The correct answer is (C). Here are two ways to find the slope:
slope (m) = 4 − 1 = 3 −3 − 2 −5 slope (m) = 1 − 4 = −3 2 − (−3) 5 −5 3 −3 5 3 5 3 Example: In the xyplane, at what point along the yaxis does the line passing through points (5,–2) and (3,4) intersect that axis? (A) –8 (B) (C) (D) (E) Solution: The correct answer is (E). The question asks for the line’s yintercept (the value of b in the general equation y = mx + b). First, determine the line’s slope:
slope (m) = y2 − y1 4 − (−2) 6 = = = −3 x 2 − x1 3− 5 −2 –2 3 7 13 5 In the general equation (y = mx + b), m = –3. To find the value of b, substitute either (x,y) value pair for x and y, then solve for b. Substituting the (x,y) pair (3,4):
y = −3x + b 4 = −3(3) + b 4 = −9 + b 13 = b Example: In the xyplane, the (x,y) pairs (0,2) and (2,0) define a line, and the (x,y) pairs (–2,–1) and (2,1) define another line. At which of the following (x,y) points do the two lines intersect? (A) ( 43 , 23 ) (B) (C) (D) (E) (,) (− , ) ( ,− ) (− , − )
34 23 13 22 3 4 2 3 3 4 2 3 www.petersons.com Additional Geometry Topics, Data Analysis, and Probability 285 Solution: The correct answer is (A). For each line, formulate its equation by determining slope (m), then yintercept (b). For the pairs (0,2) and (2,0):
y= ( )x + b
0− 2 2− 0 (slope = −1) 0 = −2 + b 2=b The equation for the line is y = –x + 2. For the pairs (–2,–1) and (2,1):
y= ( 1− ( −1) 2− ( −2 ) ) x + b (slope = 1 ) 2 1 = 1 (2) + b 2 0=b The equation for the line is y = substitution. For example:
1 2 3 2 1 2 x. To find the point of intersection, solve for x and y by x = −x + 2 x=2 x= y=
4 3 2 3 The point of intersection is defined by the coordinate pair ( 4 , 2 ). 33 Example: Referring to the xyplane above, if the scales on both axes are the same, which of the follo...
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