Linear Equations in Two Variables

Transcription

1 Section. Linear Equations in Two Variables Section. Linear Equations in Two Variables You should know the following important facts about lines. The graph of b is a straight line. It is called a linear equation in two variables. a) The slope steepness) is m. b) The -intercept is 0, b. The slope of the line through, and, is change in rise change in run. a) If m > 0, the line rises from left to right. b) If 0, the line is horizontal. c) If m < 0, the line falls from left to right. d) If m is undefined, the line is vertical. Equations of Lines a) Slope-Intercept Form: b b) Point-Slope Form: m c) Two-Point Form: d) General Form: e) Vertical Line: a f) Horizontal Line: b A B C 0 Given two distinct nonvertical lines L : m b and L : m b a) L is parallel to L if and onl if m m and b b. b) L is perpendicular to L if and onl if m m. Vocabular Check. linear 7. a) A B C 0 iii) general form. slope b) a i) vertical line. parallel c) b v) horizontal line. perpendicular d) b ii) slope-intercept form. rate or rate of change e) m iv) point-slope form. linear etrapolation. a). Since the slope is positive, the line rises.. a) 0. The line is horizontal. Matches L. Matches L. b). Because the slope is negative, the line falls. b) m is undefined. The line is vertical. Matches L. Matches L. c). The line falls. Matches L. c). Because the slope is positive, the line rises. Matches L.

2 Chapter Functions and Their Graphs.., ) m = 0 m = undefined m = m = m = m =, ) m =. Two points on the line: 0, 0 and,. The line appears to go through, 0 and,. Slope rise run Slope 0 7. Two points on the line: Slope rise run 0, and, 0. The line appears to go through 0, 7 and 7, 0. Slope Slope: -intercept: 0, Slope: -intercept: 0, 0 Slope: -intercept: 0, 0, ) 0 7 0, ) 0 0, 0) Slope:, vertical line Slope: undefined -intercept: 0, No -intercept Slope: 0 0, ) 7 -intercept: 0, ) 0,

3 Section. Linear Equations in Two Variables Slope: 7 -intercept: 0, 0, ) Slope: -intercept: 0, 0, ) 0, horizontal line Slope: 0 -intercept: 0, 0, ) Slope: 0 -intercept: 0, 0, ) 0 0. Slope: undefined vertical line) No -intercept 7 0 Slope: undefined vertical line) -intercept: none.. Slope, ). m is undefined., ) 0, ), ), ), ). Slope 0 0 0, 0) 0 0, 0). 7,,

4 Chapter Functions and Their Graphs. Slope 7 7,, Slope ,.).,.).7,.) 9.,.) 9. Point:,, Slope: 0 0. Point:,, Slope is undefined. Since 0, does not change. Three points are 0,,,, and,. Because m is undefined, does not change. Three other points are:, 0,,,,.. Point:,, Slope:. Point: 0,, Slope: Since, increases b for ever one unit increase in. Three points are,, 7,, and,. Because, decreases b for ever one unit increase in. Three other points are: 0,, 9,,, 7.. Point:,, Slope is undefined.. Point:,, Slope: 0 Since m is undefined, does not change. Three points are, 0,,, and,. Because 0, does not change. Three other points are:,,,, 0,.. Point:,, Slope:. Point: 0, 9, Slope: Since, increases b for ever one unit increase in. Three additional points are,,,, and, 0. Because, decreases b for ever one unit increase in. Three other points are:,,,,,. 7. Point: 7,, Slope:. Point:,, Slope: Since, increases b unit for ever two unit increase in. Three additional points are 9,,, 0, and,. Because, decreases b for ever unit increase in. Three other points are:,,, 7,, Point 0, ; 0. Point 0, 0 ; 0 0, ) , 0). Point, ;. Point 0, 0 ;, ) 0 0 0, 0)

5 Section. Linear Equations in Two Variables. Point, 0 ;. Point, ; 0, 0) 0 7, ). Point, ; m is undefined.. Point 0, ; m is undefined. The line is vertical., ) Because the slope is undefined, the line is a vertical line passing through 0, which is the equation. 0, ) 7. Point, ; 0 The line is horizontal. ), ). Point, ; 0 0 0, 9. Point.,. ; 0. Point.,. ; ,.) ,.) 0., and,., ), ),,, 7 7 7, ), ) 7

6 Chapter Functions and Their Graphs., and, 7. Since both points have, the slope is undefined, and the line is vertical., 7), ) 0,,, 0 0, ), )., and,.,,,, ),, ) ), 7. 0, and 9 0, , 0 9 9, 0,,, ,, , 0. and, 0. 0., 0.,, , 0.), 0.) , 0.) 0 or 0..,.)

7 Section. Linear Equations in Two Variables 7., and 0 The line is horizontal.,.,, ) ) ),,, 0 0, ),. 7, and 7, The line is vertical. 9 and is undefined. 0 ) 7, ) 7 ).,,., The slope is undefined. The line is vertical.., 0.)., ) 7 7, ).. L : 0,,, 9 Slope of L : L : 0,,, Slope of L : L and are perpendicular. L. L :,,, m L :,,, m The lines are neither parallel nor perpendicular. 7. L :,,, 0 Slope of L : L : 0,,, 7 7 Slope of L 0 : L and 0 are parallel. L. L :,,, 9. m L :,,, m The lines are perpendicular Slope: a),, b),, Slope: a) 7,, b),,

8 Chapter Functions and Their Graphs Slope: a), 0 and 0 a) b), 7, 7, 7, 7 Slope: a), 7, b) 0, 0, m is undefined. 7 7 b) 0, 7, Slope: 0 a) 0,, 0 0 b) The reciprocal of 0 is undefined. The line is vertical, passing through,. m is undefined. a),, m is undefined. The line is vertical, passing through,. b),, Slope: undefined a) The original line is the vertical line through. The line parallel to this line containing, is the vertical line. b) A perpendicular to a vertical line is a horizontal line, whose slope is 0. The horizontal line containing, is the line. Slope: a).,.,... b).,., Slope: a).9,., b).9,.,

9 Section. Linear Equations in Two Variables , 0, 0,. 0 0., 0, 0,. 0 c c, c 0. d, 0, 0, d,, c c c 0 d d d d d 0. a) b) a). a) c) b) c) c) b) c) a) 9 9 b) and c) are perpendicular. a) is parallel to c). b) is perpendicular to a) and c). b) 7. a) b) c) 0 a) and b) are parallel. c) is perpendicular to a) and b). c) b) a). a) b) c) a) is parallel to b). c) is perpendicular to a) and b). 0 b) a) 0 c) 9. Set the distance between, and, equal to the distance between, and,., ) This line is the perpendicular bisector of the line segment connecting, and,., ), )

10 0 Chapter Functions and Their Graphs 90. Set the distance between, and, equal to the distance between, and, , ) 0 7,, ) 9. Set the distance between, and, equal to the distance between 7, and, This line is the perpendicular bisector of the line segment connecting, and 7,., 7 7, ) 9. Set the distance between, and, equal to the distance between and, ,,, 7,, 9. a). The sales are increasing units per ear. 9. a) 00. The revenues are increasing 00 units per da. b) 0. There is no change in sales during the ear. c) 0. The sales are decreasing 0 units per ear. b) 00. The revenues are increasing 00 units per da. c) 0. There is no change in revenue during the da. Revenue remains constant.) 9. a) 0,,7,,,7 :,,7,,,99 :,,99,, 9,77 : The average salar increased the most from 990 to 99 and the least from 99 to 99. CONTINUED,7,7 0,99,7 9,77,99 0., 9,77,, 7,0 :, 7,0, 0, 79,9 : 0, 79,9,,,9 : 7,0 9,77 79,9 7,0 0,9 79,

11 Section. Linear Equations in Two Variables 9. CONTINUED b) 0,,7,,,9 :,9,7 0 $. c) The average salar for senior high school principals increased b $. per ear over the ears between 990 and a) The greatest increase of $. million is between 00 and 00. The least increase of $. million is between 000 and 00. b) Slope c) Each ear the net profit increases b $9. million feet a) and b) c) Vertical measurements Horizontal measurements d) Since, for ever change in the horizontal measurement of units, the vertical measurement decreases b. e) % grade 99., 0, 00. V 0 t V 0 t V t, t 0,,.0 V.0 t V.0t. V.t., t 0 0. Matches graph b). The slope is 0, which represents the decrease in the amount of the loan each week. The -intercept is 0, 00, which represents the original amount of the loan. 0. Matches graph c). The slope is, which represents the increase in the hourl wage for each unit produced. The -intercept is 0,., which represents the hourl rate if the emploee produces no units. 0. Matches graph a). The slope is 0., which represents the increase in travel cost for each mile driven. The -intercept is 0, 0, which represents the fied cost of $0 per da for meals. This amount does not depend on the number of miles driven. 0. Matches graph d). The slope is 00, which represents the amount b which the computer depreciates each ear. The -intercept is 0, 70, which represents the original purchase price.

12 Chapter Functions and Their Graphs 0., 0.,,.0 : t 9 represents 999, 9, t t represents 00,, t For 00, use t : $. N 79.t 0,. For 00, use t 0: 0 $7. t represents 00: N 79. 0,. 9. stores t 0 represents 00: N ,.. stores These answers are not reasonable because the are negative. 07. Using the points 0, 7 and, 0, where the 0. 0,,000 and 0, 000 first coordinate represents the ear t and the second coordinate represents the value V, we have V 00t,000, 0 t 0 0 V 7t 7, 0 t. 09. a) 0, 0,7,,,9 : b) For 00, use t :,007 students.,9 0, t 0,7 79. For 00, use t 0: 0, students. c) The slope is 79., which represents the increase in the number of students each ear. 0. a) Average annual salar change from 990 to 00: c),7, 0, 9 students per ear 9, b,, so N t 9t,. The slope, 9, represents the average annual change in enrollment. b) Using a) to estimate the enrollment in: 99:, 9 0,7 students 99:, 9,00 students 00:, 9 7,79 students d) Answers will var.. Sale price List price % of the list price. W S L 0.L S 0.L. a) c) C,00.t.0t.7t,00 P R C 7t.7t,00 0.t,00 b) R 7t d) 0 0.t,00,00 0.t t hours

13 Section. Linear Equations in Two Variables. 0, 0 and, 7 a) b) c) p 0 0 p p units 9 9 units. a) b) c) m 0 0 m d) Since, each -meter increase in will increase b meters.. W 0.07S C Median salar in thousands of dollars) Using a calculator, the linear regression line is 00.t 7.. Choosing the points 7, 0 and 0, 00 : 0 Year 99) t 7 0.t.. The answer varies depending on the points chosen to estimate the line. t 9. a) and b) Cellular phone subscribers in millions) Year 0 990) c) Answers will var. Find two points on our line and then find the equation of the line through our points. Sample answer:.7.0 d) Answers will var. Sample answer: The -intercept should represent the number of initial subscribers. In this case, since b is negative, it cannot be interpreted as such. The slope of.7 represents the increase in the number of subscribers per ear in millions). e) The model is a fairl good fit to the data. f) Answers will var. Sample answer: million subscribers in a) and b) c) Two approimate points on the line are 0, and 9, 9. d) Average test score Average quiz score e) Each point will shift four units upward, so the best-fitting line will move four units upward. The slope remains the same, as the new line is parallel to the old, but the -intercept becomes 0, 9 0, 9 so the new equation is 9 9.

14 Chapter Functions and Their Graphs. False. The slope with the greatest magnitude corresponds to the steepest line.. Using the Distance Formula, we have AB, BC 0, and AC. Since 0, the triangle is a right triangle.., and, : m 7 0, and 7, 7 : m False, the lines are not parallel On a vertical line, all the points have the same -value, so when ou evaluate, ou would have a zero in the denominator, and division b zero is undefined.. No. The slope cannot be determined without knowing the scale on the -ais. The slopes will be the same if the scale on the -ais of a) is and the scale on the -ais of b) is. Then the slope of both is.. Since >, the steeper line is the one with a slope of. The slope with the greatest magnitude corresponds to the steepest line. 7. The V-intercept measures the initial cost and the slope measures annual depreciation.. No, the slopes of two perpendicular lines have opposite signs. Assume that neither line is vertical or horizontal.) 9. is a linear equation with slope and -intercept 0,. Matches graph d). 0. Intercepts:, 0, 0, Matches graph c).. is a quadratic equation. Its graph is a. parabola with verte, and -intercept 0,. Matches graph a). Intercepts:, 0,, 0, 0, Matches graph b) or or 7 0 b ± b ac a ± ± ± No real solution The square root of 9 cannot be negative Answers will var.