Chapter 7 Linear Equations and Graphs 7.1 Slope-Intercept Form 1. a) m = 1, y-intercept: 2

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1 Chapter 7 Linear Equations and Graphs 7.1 Slope-Intercept Form 1. a) m, -intercept: b) m =, -intercept: c) m, -intercept: d) m =.7, -intercept:.. a) + = 7 = + 7 m = 1, -intercept: 7 b) + + = + 1 m =, -intercept: 1 c) + = + 1 = = = + m =, -intercept: d) + 1 = + 1 = _ 1 = 1 = m, -intercept:. a) = 1 b) = c) = d) =. e) = + 1 f ) = +. a) b) c) d) e) f ) MHR Answers

2 . a) m, b, = + 1 b) m = 1, b =, = + c) m =, b =, = d) m =, b =, = + e) m =., b =, =. f) m =, b =, = +. a) Replace with 1 and with 8 in the equation + b. + b 8 (1) + b Solve for b. 8 = + b 8 = + b = b b) Replace with and with 1 in the equation + b. + b 1 ( ) + b Solve for b. 1 _ = + b 1 + = + + b = b = b 7. a) m = b) m = 8. a) = b) m = ; the price per raffle ticket c) b = ; the cost of the pair of hocke tickets d) 7 tickets 9. a) = 7 b) $ loss; $ profit; $17 profit c) 8 competitors 1. a) $ b) + c) b =, which represents the fied charge d) discrete because rental is charged per whole hour 11. a) Amount of Fuel (L) Distance Travelled (km) 8 b) Substitute two points on the line, for eample, (, ) and (, ), into the slope formula: m = 1 1 _ m = m = _ m = _ The line intersects the -ais at, so b =. c) Replace m with _ and b with in the slope-intercept form: = _ +. d) The amount of fuel in the car s tank before driving an distance. e) The tank is empt when =. Replace with and solve for : = _ + = _ + = _ + ( = _ )( ) = ( )( _ 7 = ) The tank will be empt after 7 km Mathematics 1 Eercise and Homework Book MHR 8

3 1. a) b) Cost of Printing ($) Cost of Number of Printing Programs ($) Number of Programs 8 c) m ; the cost of printing each program d) b = ; the fied cost e) + f ) programs 1. a) b) Use two points from the table to find the slope: (, 8) and ( 1, 1). m = m = 1 m = m = 9 c) b =, which represents the temperature in Fahrenheit when the temperature is C. d) Replace m with 9 and b with in the slope-intercept form: = 9 +. e) In the equation = 9 +, replace with and with. Then, solve for. = 9 + = 9 + = 9 9 ( ) = ( 9 )( 9 ) 9 _ 1 9 = f) Use the = 9 + form of the equation, replacing with. = 9 + = 9 ( ) + = 7 + = C = F Use the = 9 1 _ 9 form of the equation, replacing with 1. = 9 (1) 1 _ 9 = F = 7.8 C Use the = 9 + form of the equation, replacing with. = 9 + = 9 () + = C = F 1. 8 people 1. a) Eample: MHR Answers

4 b) The slope is and the -intercept is (, 1). 1 b) (, ) c) (, ), no -intercept c) Eample: I prefer using the slope and -intercept. There is less computation involved when writing the equation in slope-intercept form. It is simple to graph the line using the slope and the -intercept. 7. General Form 1. a) + 1 = b) + 7 = c) 8 1 = d) =. a) (, ), (, 8) - - d) no -intercept, (, 1 _ ) Mathematics 1 Eercise and Homework Book MHR 8

5 . a) domain: R; range: R To find the slope, use the points (, ) and (, ). m = 1 1 _ m = m = _ The -intercept is (, ) and the -intercept is (, ). In the slope-intercept form, replace m with _ and b with : = _ +. Multipl both sides of the equation b : = ( + ) = + 1 Bring all terms to one side of the equation: = = = b) domain: R; range: = Since this is a horizontal line, the slope is. The -intercept is (, ). The equation of the line in general form is + =.. Eamples: a) + = b) + = c) 8 = d) 7 = e) + =. a) B = 1 b) A = c) C =. a) Let 1 represent the number of calories burned swimming for minutes. Let 1 represent the number of calories burned biking for minutes. The equation to represent the total number of calories burned is =. b) To find the -intercept, replace with = 1 + 1() = 1 = = The -intercept represents how man minutes he would need to spend swimming if he burned calories b onl swimming. To find the -intercept, replace with = 1() + 1 = 1 = = The -intercept represents how man minutes he would need to spend biking if he did not spend an time swimming. c) The domain is. There can be no values less than or greater than. The range is. There can be no values less than or greater than. d) Replace with 1 and solve for = 1 + 1(1) = = = 1 1 = 7 97 He would need to swim 197 minutes, or hours and 17 minutes. 7. a) + = b) m = ; (, ) and (, 11); domain:, range: 11 c) minutes, or hours and minutes square units 9. Eamples: a) (, 1) and (, ), 1 + = b) (, 8) and (7, 8), 8 = c) (, 1) and (, ), = d) (., ) and (., ), + 7 = 8 MHR Answers

6 7. Slope-Point Form 1. a) m =, (, 7) b) m, (, ) c) m =, (, ) d) m, (, 1). a) = + 11 ; + 11 = b) = ; + + = c) = ; 1 = d) = + 19; + 19 =. a) For slope-point form, replace m with and ( 1, 1 ) with ( 1, ): 1 = m( 1 ) ( + ) = ( + 1) For slope-intercept form, rewrite ( + ) = ( + 1) in the form = m + b: ( + ) = ( + 1) + = + + = + = 11 For general form, rewrite = 11 in the form A + B + C = : = ( 11 ) = 11 = 11 = 11 b) Slope-point form: ( + ) ( + 1 ) Slope-intercept form: = General form: = c) Slope-point form: ( ) = 1.( 1) Slope-intercept form: = General form: + 1 d) Slope-point form: First find the slope of the line through the points (, 8) and ( 7, 9). m = 1 1 _ 9 ( 8) m = 7 ( ) m Then, replace m with 1 and ( 1, 1 ) with (, 8) in the slope-point form. 1 = m ( 1 ) ( + 8) ( + ) Slope-intercept form: 11 General form: 11 = ( 1 11 ) = 11 = 11 = 11 e) Slope-point form: ( + ) ( + 1) Slope-intercept form: General form: =. Eamples: a) ( ) b) = 1( ) c) + = ( ). a) 1 = ( + ); 1 = b) 8 = ( + 1); + 1 = c) The slope of the line + 1 is. So, the second line must have this same slope. Since it passes through the point ( 1, ), the equation of the second line in slope-point form is = ( + 1). To convert this to general form, move all terms to the left side of the equation: = ( + 1) = ()( ) = () ( ) 8 = 8 + = + = + = + + = d) + = ( ); + + = e) = ( ); + 1 = f) = ( ); + = Mathematics 1 Eercise and Homework Book MHR 87

7 . Write the equation of the line with -intercept (1, ) and -intercept (, ). Find the slope. 7. m = 1 1 m = 1 m = 1 m Substitute m and b = into the slope-intercept form:. Replace with and with. ( ) = 1 = Since replacing with and with in the equation of the line results in a true statement, the point (, ) is on the line. A - - B - D C AC: = BD: = 8. The -intercept is (, ) and the -intercept is (, ) k 1. a) Lines and b) Line c) Lines 1 and 11. a) Cost ($) 8 1 Distance (mi) b) m =, which represents the cost of operating a snowmobile per mile c) b = ; the fied cost of operating a snowmobile d) + 1 = e) $ 1. a) = b). cm/h; at 1 hours it was 1 cm tall c) rate of burn per hour d) the height at 1 hours 1. Rewrite each equation in slope-intercept form. Enter the equations into a graphing calculator or graphing program and read the intersection points, which are the vertices. Line 1: + 18 = + 18 = 18 = = + 18 = + 18 _ = + 18 _ = + Line : = = + 7 = = 7 = 7-88 MHR Answers

8 Line : 1 = 1 = 1 = = + 1 = + 1 _ = _ + 1 _ = 7 Enter the equations into a graphing calculator or graphing program. (-, 8) 8 Therefore, the -intercept is (, ). Verif b replacing with and solving for to find the -intercept. = Therefore, the -intercept is (, ). The above shows that the prediction was correct. 1. a) Eample: 1. = 1 9 ( 1) b) in the ear (, ) 8 7. Parallel and Perpendicular Lines 1. a) perpendicular b) parallel c) perpendicular d) perpendicular e) perpendicular f ) parallel. a) parallel: ; perpendicular: 1 b) parallel: 1; perpendicular: (, -7) The vertices of the triangle are (, 8), (, 7), and (, ). 1. a) + = b) The -intercept is ( 8, ) and the -intercept is (, ). The denominator of the -term in the original equation is the -intercept. The denominator of the -term in the original equation is the -intercept. c) Eample: Predict that the -intercept is (, ) and the -intercept is (, ). Verif b replacing with and solving for to find the -intercept. = c) parallel: ; perpendicular: 1 d) parallel: ; perpendicular: undefined e) parallel: ; perpendicular:. a) n = b) n = c) n =. d) n =. a) r = b) r c) r = 18 d) r = 8. a) + 7 = b) + + = c) + =. a) + = b) + 7 = c) 1 = 7. a) 1, 1 b) no, the equations represent the same line 8. a) 1 = b) + 1 = 9. 1 = 1. a) The slope of side MN is, the slope of side NC is 7, and the slope of side MC is. Since no two slopes have a product of 1, these points do not represent the vertices of a right triangle Mathematics 1 Eercise and Homework Book MHR 89

9 b) The slope of DF is 1, the slope of FG is 7, and the slope of DG is. Since the product of the slopes of sides DF and DG is 1, the points represent the vertices of a right triangle. 11. Eamples: a) Find the slope of + 11 = b writing the equation in slope-intercept form, = m + b: + 11 = + 11 = 11 = = + 11 = + 11 The slope is. An line with the same slope but a different -intercept will be parallel to the given line. Eample, = + 1. An infinite number of equations can be written in the form = m + b, where m = and b 11. b) A line perpendicular to + 11 = has a slope that is the negative reciprocal of, or 1. Eample,. An line with a slope of 1 will be perpendicular to the given line, regardless of the -intercept. There is an infinite number of these lines = 1. The line must have a slope of 1 and a -intercept of k = k = 1. k = a) Write the equation in slope-intercept form: k 1 = k k 1 = k 1 = k = k + 1 = k + 1 _ = k _ + _ 1 = _ k 1 Therefore, the slope of the first line is k. 8 k + = 8 8 k + = 8 k + = 8 k + = 8 k = 8 _ k k = _ 8 k k = 8 k + k The slope of the second line is 8 k. Since the lines are parallel, the slopes must be equal. Set the two slopes equal to each other and solve for k. k = 8 k k _ k = 1 k = ± The values of k are and. b) Since the lines are perpendicular, the slope must have a product of 1. ( k )( 8 k ) = 1 8k k = 1 1 Therefore, there are no values of k that will work. 18. a) 1 = b) + = 9 MHR Answers

10 19. Eample: =, =, =, = c) m =.; b = ; the slope represents the redemption value per jug; the -intercept represents the amount of mone alread raised d) 1 jugs 7. General Form. a) = b) + = c) + + =. a) -intercept is (, ); -intercept is (, ) Chapter 7 Review 7.1 Slope-Intercept Form 1. a) m = ; b = ; = b) m = 1; b = ; = + c) m = 1 1 ; b ; = + 1 d) m = ; b = ; =. b = 8. If the line = m 8 passes through the point (, ), replace with and with. Solve for m. = m 8 = m( ) 8 = m = m = m 1 = m 7 = m. a) =. + b) Mone Raised ($) b) -intercept is (, ); -intercept is (, ) a) + 1 = b) -intercept is (7, ), -intercept is (, ); The -intercept represents the number of das of growth under ideal conditions to reach a height of m. The -intercept represents the number of das of growth under less than ideal condition to reach a height of m. c) 7, d) 9 das Number of Jugs Mathematics 1 Eercise and Homework Book MHR 91

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