Chapter 4 ( 2, 2 ). Chapter 4 Opener. Section 4.1. Big Ideas Math Blue WorkedOut Solutions. Try It Yourself (p. 141) = = = = 15. The solution checks.


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1 Chapter Chapter Opener Tr It Yourself (p. ). ab = ( ) = ( ) = = = =. a b = ( ). a b = = = = + = + = + 9 =, or. a ( b a ). Point Q is on the ais, units up from the origin. So, the coordinate is, and the coordinate is. The ordered pair (, ) corresponds to point Q.. Point P is units to the right of the origin and units up. So, the coordinate is, and the coordinate is. The ordered pair (, ) corresponds to point P.. Start at the origin. Move units left. Point R is located,. at. Point N is located in Quadrant II. Section.. Activit (pp. ). a. Sample answer: Solution Points c. Sample answer: d. Sample answer: Check (, ). = +? = +? = + = The solution checks. e. es; Because the line is the graph of the equation, all points on the line are solution points. f. Sample answer: Solution Points = + + = Solution Points + = = + ( ) + = () + = Solution Points = + () + = O O = + ( ) + = + = b. Sample answer: The ordered pairs are (, ) and (, ). Each point lies on the line. g. es; The graph of the equation is the set of all solutions of the equation. So, each of these solutions falls on the line. h. The equation = a + bis a linear equation because its graph is a line. Copright Big Ideas Learning, LLC 9
2 Chapter. d. In the second graph, it is easier to see where the line crosses the ais and the ais.. A linear equation has the form = a + b. You can draw the graph of a linear equation b finding an two solution points, plotting the points, and drawing a line through the points. Sample answer: The equation = + is a linear equation. The equation = + is not a linear equation.. a. Yes, the line crosses the ais at = ; No, ou cannot tell where the line crosses the ais. b. Because the line crosses the ais at =, ou can adjust the minimum value of to be or lower.. Sample answer: You should use a graphing calculator. You will get a more accurate graph than if ou graph it b hand because of the decimals.. On Your Own (p. ). = (, ) = ( ) (, ) = (, ) =. The graph of = is a vertical line passing through (, ).. The graph of =. is a horizontal line passing. (, ) O through (,. ). = (,.) =. = + (, ). =.( ) + (, ) =. + (, ) =. + (, ) =.( ) + (, ) 9. = + (, ) = + (, ) = + (, ) = + From the graph, ou can see that = when =. So, the storm becomes a hurricane hours after it enters the Gulf of Meico.. Eercises (pp. ) Vocabular and Concept Check. The solutions of the equation = + can be represented b a line.. The equation = + does not belong because it is not of the form = a + b. 9 Copright Big Ideas Learning, LLC
3 Chapter Practice and Problem Solving. Sample answer: = = () = = (, ). Sample answer: = + ( ) + = () + =. The graph of = (, ). is a horizontal line passing through. The graph of = is a vertical line passing through (, ). (, ) = = (, ) O (, ) = + 9. = (, ) = (, ) = (, ). = (, ) = (, ) = (, ) = =. = (, ) = (, ) = () (, ). = ( ) = (, ), = = (, ) = Copright Big Ideas Learning, LLC 9
4 Chapter. = + (, ) = + (, ) = + (, ). The graph of =. is a vertical line passing through (., ). (., ) =.. = + ( ) = (, ) =. The graph of = is a horizontal line passing through,., = (, ) =. The graph of =. is a horizontal line passing through (,. ). (,.) =. (, ) =. The graph of,. = is a vertical line passing through. The graph of the line is horizontal, not vertical, and passes through (, ).. The graph of = is a horizontal line passing through (, ). 9. a. O (, ) (, ) = = = The graph tells ou that the total cost for tet messaging is $ no matter how man tet messages ou send. = + (, ) = + (, ) = () + (, ) = + 9 Copright Big Ideas Learning, LLC
5 Chapter b. From the graph, ou can see that when =.,. So, it costs about $ to mail the package. c. = + =. + =. + =. The cost of mailing the package is $... = = + = + (, ) = ( ) + (, ) = + (, ) = +. + = = + = + = 9 9 ( ) = (, ) = ( ) 9 9 (, 9) = 9 (, ) = 9. + = = + = + = + (, ) = + (, ) = + = + (, ). +. =.. = +. = + = + (, ) = + (, ) = + (, ) = +. a. Your beginning balance is $. After months, ou have +. = + = $. Plot, to graph the linear equation. (, ) and Balance (dollars) Number of months b. From the graph, ou can see that = when =. So, it will take ou months to save enough mone to bu acres of land on Mars. Copright Big Ideas Learning, LLC 9
6 Chapter. a. Sample answer: n S = ( n ) S ( n, S) 9 S = ( ) (, ) S = ( ) (, ) S = ( ) (, ) S = ( ) (, ) S (, ) (, ) (, ) (, ) n es; The graph of the equation is a line. b. no; The value n =. does not make sense because n is the number of sides of the polgon, so it must be an integer.. Sample answer: For a earold student, let =. = = = So, the sea level rose millimeters since the student was born. Let be the amount the sea level rises and be the number of ears. An equation is =. = (, ) = () (, ) = (, ). a. Number of equals maimum minus pictures camera pictures can store with camera videos can hold times number of seconds of video. Let be the number of pictures and be the number of seconds of video. An equation is =. Pictures = (, ) = ( ) (, ) = ( ) (, ) = ( ) 9 (, 9 ) = ( ) (, ) Seconds of video b. You know that minute seconds equals seconds + seconds = 9 seconds. From the graph, ou can see that when = 9, =. So, our camera can store pictures after taking a 9second video. Rise in sea level (mm) = Number of ears Fair Game Review. Point A is units to the right of the origin and units up. So, the coordinate is, and the coordinate is. The ordered pair (, ) corresponds to point A. 9. Point B is units to the left of the origin and units up. So, the coordinate is, and the coordinate is. The ordered pair (, ) corresponds to point B.. Point C is units to the right of the origin and units down. So, the coordinate is, and the coordinate is., corresponds to point C. The ordered pair. Point D is units to the left of the origin and units down. So, the coordinate is, and the coordinate is., corresponds to point D. The ordered pair 9 Copright Big Ideas Learning, LLC
7 Chapter. B; females females = males m males m = m = m = There are males on the debate team. Section.. Activit (pp. 9). a. change in slope = = change in change in slope = = = change in es; All of the points are on the same line, so ou get the same slope no matter which two points ou choose. change in b. slope = = = change in change in slope = = = change in es; All of the points are on the same line, so ou get the same slope no matter which two points ou choose. change in c. slope = = change in change in slope = = = change in 9 es; All of the points are on the same line, so ou get the same slope no matter which two points ou choose. change in d. slope = = = change in change in slope = = = change in es; All of the points are on the same line, so ou get the same slope no matter which two points ou choose. b. The are similar; Sample answer: The lines BC and EF are both vertical lines and are parallel. The line that contains the points A, B, D, and E is a transversal. So, ABC is congruent to DEF because corresponding angles are congruent. The lines AC and DF are both horizontal lines and are parallel. The line that contains the points A, B, D, and E is a transversal. So, BAC is congruent to EDF because corresponding angles are congruent. Both are right triangles. So, Δ ABC and Δ DEF are similar because their angles are congruent. c.. a. BC AC EF DF = = = The ratio is for each triangle. The ratio represents the slope of the line between points A and B and between points D and E. So, the slope of the line is. d. Sample answer: It is constant. The two lines are parallel. b.. a. 9 A(, ) D(9, ) B(, ) C(, ) E(, ) F(, ) The two lines are parallel. c. Two nonvertical lines in the same plane that have the same slope are parallel. 9 Copright Big Ideas Learning, LLC 9
8 Chapter d.. = = The slope is.. = The slope is undefined. e. The lines form a right angle. The product of the slopes of the two lines is =. The lines form a right angle. The product of the slopes of the two lines is. Two lines in the same plane whose slopes have a product of are perpendicular.. The slope of a line describes the steepness of the line and whether it rises or falls from left to right.. On Your Own (pp. ). =, or. The slope is. = = The slope is.. = The slope is undefined.. The slope of ever horizontal line is because the coordinates are all the same, so the change in is. The slope of ever vertical line is undefined because the coordinates are all the same, so change in is, and division b is undefined. 9.. = The slope is. = = The slope is.. Eercises (pp. ) Vocabular and Concept Check. a. The red and blue lines rise from left to right. So, lines B and C have positive slopes. b. Line A has the steepest slope. c. no; None of the lines have an undefined slope because none of them are vertical.. Sample answer: When constructing a wheelchair ramp, ou need to know the slope.. If the slope of a line is, then the line is horizontal. Practice and Problem Solving. = The slope is.. = = The slope is... The two lines are parallel. The two lines are parallel. 9 Copright Big Ideas Learning, LLC
9 Chapter The two lines are parallel. = The slope is. =, or The slope is. =, or The slope is. = The slope is.. = ( ) The slope is.. = The slope is undefined.. = = The slope is.. = The slope is undefined.. = ( ) The slope is undefined.. =, or The slope is.. = = ( ) The slope is. 9. The order of subtraction needs to be consistent. = = The slope is.. Ramp: Hill: rise ft = run ft rise ft = run ft Because is greater than, the ramp is steeper. So, it is more difficult to walk up the ramp.. = = The slope is..... = The slope is. =, or The slope is. =, or The slope is. rise ft = run ft The pitch of the roof is.. = = The slope is. Copright Big Ideas Learning, LLC 99
10 Chapter. a. Answer should include, but is not limited to: Students will measure a wheelchair ramp in their school or neighborhood and then determine its slope. The will compare the slope to the guidelines and determine if the ramp meets the guidelines. b. Sample answer:.. m = k = k = = k = k m = k = k = = k = k ( ) ft. ft. a.... = = = The slope is. b. A slope of means that the cost increases b $ for ever miles ou drive, or the cost increases b $. for ever mile ou drive. rise ft. Boat ramp: = run ft vertical increase Road: % =. horizontal distance Because =. is greater than., the boat ramp is steeper.. es; The slope using an two of the points is, so the are on the same line. ( ) Points A and B: = = Points B and C: = = ( ) Points A and C: = = 9.. m = k = k = = k = k = k m = ( k ) = = k = k k = k =., =, = The rate of change in profit is $ per month.. Sample answer: Let, =, and let, =,. = = = = When ou switch the coordinates, the differences in the numerator and denominator are the opposite of the numbers when using the slope formula. You still get the same slope. Copright Big Ideas Learning, LLC
11 Chapter. a. Bottom of slide is inches above ground. ft in. =. ft in. So, the vertical distance of the main portion of the slide is ft. ft =. ft. The horizontal distance of the main portion of the slide is ft ft ft = ft. rise. ft slope = = =. run ft The slope of the main portion of the slide is.. b. If the bottom of the slide is inches, or foot, above the ground, the vertical distance of the main portion of the slide is ft ft = ft. rise ft slope = = =. run ft Because. >., the slope increased and the slide is steeper. Fair Game Review b. =. 9. b = b = So, b =. n = = n = n So, n =. = = =. = So, =... B; The prime factorization is, or.. Etension (pp. ) Practice. Blue line: m = = = = Red line: = = Green line: = = The slopes of the blue and red lines are. The slope of the green line is. The blue and red lines have the same slope, so the are parallel.. Blue line: = Red line: = Green line: = The slope of the blue line is. The slopes of the red and green lines are. The red and green lines have the same slope, so the are parallel.. es; The have the same slope,.. no; The are perpendicular. The line = has a slope of and the line = has an undefined slope.. es; The are both vertical lines with undefined slope.. Use the vertices of the quadrilateral to find the slope of each side. When opposite sides are parallel (have the same slope), the quadrilateral is a parallelogram. Segment AD: =, or Segment BC: =, or Segment AB: =, or ( ) Segment DC: ( ) =, or ( ) Because opposite sides of the quadrilateral do not have the same slope, the opposite sides are not parallel. So, the quadrilateral is not a parallelogram. Copright Big Ideas Learning, LLC
12 Chapter. Blue line: = = Red line: = = ( ) Green line: =,or ( ) The slope of the blue line is. The slope of the green line is. Because =, the blue and green lines are perpendicular.. Blue line: = = ( ) Red line: = = ( ) ( ) Green line: = = The slopes of the blue and red lines are. The slope of the green line is. Because =, the blue and green lines are perpendicular, and the red and green lines are perpendicular. 9. es; The line = is vertical and the line = is horizontal, so the are perpendicular.. no; The are both vertical lines with undefined slope, so the are not perpendicular.. es; The line = is horizontal and the line = is vertical, so the are perpendicular.. Use the vertices of the parallelogram to find the slope of each side. When adjacent sides are perpendicular, the parallelogram is a rectangle. Segment JK: = = Segment KL: =, or Segment LM: = = ( ) Segment MJ: =, or ( ) Segment JK is perpendicular to Segment KL because =. Segment KL is perpendicular to Segment LM because =. Segment LM is perpendicular to Segment MJ because =. Segment MJ is perpendicular to Segment JK because =. Because adjacent sides are perpendicular to each other, the parallelogram is a rectangle. Section.. Activit (pp. 9). a. proportional relationship; Sample answer: The graph is a line through the origin. b. not a proportional relationship; Sample answer: The graph does not pass through the origin. c. not a proportional relationship; Sample answer: The graph does not pass through the origin. d. proportional relationship; Sample answer: The graph is a line through the origin. e. not a proportional relationship; Sample answer: The rate of change in the table is not constant. f. proportional relationship; Sample answer: The rate of change in the table is constant. Copright Big Ideas Learning, LLC
13 Chapter. The quantities in parts (a), (d), and (f) are in a proportional relationship. rise For part (a): slope = = = ; run The value of for (, ) is. rise For part (d): slope = = = ; run The value of for (, ) is. For part (f): slope = = = = ; The value of for (, ) is. The value of is equal to the slope of the line. The value of represents the unit rate.. a. Both triangles share the same angle and each has a right angle. So, their angles are congruent making them similar. m b. = = m = m The final equation represents the general equation for two quantities and that are in a proportional relationship. c. For part (a): = For part (d): = For part (f): =. The graph of = mis a line with a slope of m that passes through the origin. The value of m affects the steepness of the line.. Sample answer: amount of gasoline purchased (gallons) and the amount spent (dollars); =. ; Amount spent (dollars) 9 9 Amount of gasoline purchased (gallons). On Your Own (p. )... The slope indicates that the unit cost is $ per gigabte. = = ( ) = The spacecraft would weigh kilograms on Earth. slope = = = = The ski lifts from slowest to fastest are person, Tbar, and person.. Eercises (pp. ) Vocabular and Concept Check. (, ). no; Sample answer: The graph of the equation does not pass through the origin. Practice and Problem Solving. no; Sample answer: The graph of the equation does not pass through the origin.. es; = ; Sample answer: The graph is a line through the origin.. es; = ; Sample answer: The rate of change in the table is constant.. no; Sample answer: The rate of change in the table is not constant.. Cost (dollars) Amount raised (dollars) (, ) (, ) Data used (gigabtes) 9 Tickets The slope indicates that each ticket costs $. Copright Big Ideas Learning, LLC
14 Chapter. a. = m = m 9 = m () An equation that represents the situation is = 9. b. The slope indicates that it costs $9 per hour to rent a kaak. c. = 9 = It costs $ to rent the kaak for hours. 9. a. the car; Sample answer: The equation for the car is =. Because is greater than, the car gets better gas mileage. b. The truck can travel = () = miles on gallons of gasoline. The car can travel = () = miles, on gallons of gasoline. So, the car can travel miles further than the truck.. a. Toenail Growth mm r. mm = r weeks week Toenails grow about. millimeter per week. Fingernail Growth Weeks Fingernail Growth (millimeters) change in growth. mm = change in time week Fingernails grow about. millimeter per week and toenails grow about. millimeter per week. So, fingernails grow faster than toenails. b. Growth (millimeters) Fingernails Toenails Time (weeks) The graph that represents fingernails is steeper than the graph that represents toenails. So, fingernails grow faster than toenails Because and are in a proportional relationship, the equation that relates them is = m, where m is the slope. Rewriting the equation produces m. = So, the ratio of to is equal to the slope of the line.. a. es; The line passes through the origin. change in b. =. change in An equation of the line is =.. The slope indicates that the temperature decreases b. F for each foot increase in altitude. c. The altitude is feet, so find the temperature change when =.. =.(.) = 9. The temperature at the top of the mountain is 9. F cooler than the bottom of the mountain. So, the temperature at the top of the mountain is F 9. F =. F.. a. es; The equation is d =, t which represents a proportional relationship. Distance (feet) d 9 t Time (seconds) b. es; The equation is d = r, which represents a proportional relationship. Distance (feet) d 9 r Rate (feet per second) c. no; The equation is t =, which does not r represent a proportional relationship. Time (seconds) t 9 r Rate (feet per second) Copright Big Ideas Learning, LLC
15 Chapter d. part c; It is called inverse variation because when the rate increases, the time decreases, and when the rate decreases, the time increases. Fair Game Review. = (, ) = (, ) = = (, ) = 9 + = =. B; S ( n ) = ( ) Stud Help Available at BigIdeasMath.com. Quiz... = + (, ) = ( ) + (, ) = + (, ) = +. = (, ) = = = 9 9,,. = (, ) = = (, ) (, ). = (, ) = = =,, =. The graph of =. The graph of =. is a vertical line passing is a horizontal line,.,.. through ( ) passing through = O =. Copright Big Ideas Learning, LLC
16 Chapter. = = The slope is.. = = The slope is.. = The slope is undefined.. = = ( ) The slope is. 9. The slope of a line parallel to the line in Eercise is because parallel lines have the same slope. The slope of a line perpendicular to the line in Eercise is because ( ) =.. The lines are not parallel because the do not have the same slope. The lines are perpendicular because = is a horizontal line and = is a vertical line.. Balance at Number Your Amount = beginning of times balance withdrawn of month ATM used ATM fee Number of times ATM used Let be our balance and be the number of times the ATM was used. = = An equation is =. = (, ) = ( ) (, ) = () (, ). The slope indicates that ou take hours of cello lessons per week.. a. = m = m. = m An equation that represents the situation is =.. b. The slope indicates that the unit cost is $. per guest. c. =.( ) = It would cost $ to provide food for ten guests. Section.. Activit (pp. ). a. =,or The slope is = + (, ) = + (, ) = + (, ) = +. The line crosses the ais at (, ). 9 = 9 Copright Big Ideas Learning, LLC
17 Chapter b. = + (, ) = ( ) + (, ) = + (, ) e. = + (, ) = + (, ) = + (, ) c. = + = = The slope is. The line crosses the ais at (, ). = (, ) = ( ) = = (, ) (, ) = = The slope is. The line crosses the ais at (, ). f. = = The slope is. The line crosses the ais at (, ). = = The slope is. The line crosses the ais is at,. = + = (, ) = = (, ) = (, ) d. = + (, ) = + (, ) = + (, ) g. = (, ) = ( ) (, ) = (, ) = + = The slope is. The line crosses the ais at (, ). = = The slope is. The line crosses the ais at (, ). Copright Big Ideas Learning, LLC
18 Chapter h. = =, or = (, ) ( ) = (, ) = (, ) a. Equation = + Slope of Graph b. = + c. = d. = + e. Point of Intersection with ais (, ) (, ) (, ) (, ) = + (, ) = (, ) f. i. j. The slope is (, ).. The line crosses the ais at = + (, ) = ( ) + (, ) = + (, ) = + ( ) = = ( ) The slope is. The line crosses the ais at (, ). = (, ) = (, ) = () (, ) = g. h. = = i. (, ) (, ) = + (, ) j. = (, ) k. es; es; Sample answer: The slope of the graph is the coefficient of the term in the equation. The coordinate of the point of intersection with the ais is the constant term in the equation.. a. no; Sample answer: None of the graphs of the lines pass through the origin. b. = m b = m b = m b = m b = m = m + b c. m represents the slope; b represents the coordinate of the point of intersection with the ais. = = The slope is. The line crosses the ais at (, ). Copright Big Ideas Learning, LLC
19 Chapter. The graph of the equation = m + b is a line with slope m that crosses the ais at (, b ). a. The value of m represents the slope. So, the value of m affects the steepness of the line, and whether it rises or falls from left to right. b. The value of b represents the coordinate of the point of intersection with the ais. So, the value of b affects where the graph of the equation crosses the ais. c. Answer should include, but is not limited to: Students will choose three linear equations (not in the table in Activit ), graph their equations, find the slopes of the lines, and find the intercepts. The will then compare their results to the general statements in parts (a) and (b).. The equation = m + b is called the slopeintercept form of the equation of a line because m is the slope and b is the intercept. Answers will var.. On Your Own (pp. 9). = = +. The slope is, and the intercept is. = = + The slope is, and the intercept is.. For the equation =, the slope is and the Plot intercept is.,. Use the slope to find another point on the line. rise run Plot the point that is unit right and unit up from,. Draw a line through the two points. The line crosses the ais at (, ). So, the intercept is. =. For the equation = +, the slope is and the intercept is. Plot (, ). Use the slope to find another point on the line. rise run Plot the point that is units right and unit down from (, ). Draw a line through the two points. The line crosses the ais at (, ). So, the intercept is.. The slope is. So, the cost per mile is $. The intercept is.. So, there is an initial fee of $. to take the tai.. Eercises (pp. ) Vocabular and Concept Check. To find the intercept of the graph of + =, find the coordinate of the point where the graph crosses the ais.. es; The equation = is equivalent to the equation = +. The slope is and the intercept is.. Sample answer: You are saving to bu an MP plaer. You have $ saved and plan to save $ per week. The equation = + represents this situation, where represents the number of weeks and represents the total amount saved in dollars. The slope is. So, the amount saved per week is $. The intercept is. So, ou alread have $ saved. Practice and Problem Solving. B; The slope is, and the intercept is.. A; The slope is,. C; The slope is. = + = = + and the intercept is., and the intercept is. The slope is, and the intercept is. Copright Big Ideas Learning, LLC 9
20 Chapter. = + The slope is, and the intercept is. 9. = = + The slope is, and the intercept is.. =. + The slope is., and the intercept is... + = = = + The slope is, = = + The slope is,.. = = +.. and the intercept is. and the intercept is. The slope is, and the intercept is.. = = + The slope is ( ), and the intercept is.. a. The slope is and the intercept is. Plot (, ). Use the slope to find another point on the line. rise = = run Plot the point that is units right and units down from (, ). Draw a line through the two points. Height (feet) b. From the graph, the line crosses the ais at (, ). The intercept is. So, the skdiver lands on the ground after seconds. The slope is. So, the skdiver falls to the ground at a rate of feet per second.. For the equation =, + the slope is and the intercept is. Plot (, ). Use the slope to find another point on the line. rise run Plot the point that is units right and unit up from (, ). Draw a line through the two points. The line crosses the ais at (, ). is. = = + Time (seconds) + So, the intercept. = +. =. + The slope is., and the intercept is.. The intercept should be. Rewrite the equation as = + ( ). So, the slope is and the intercept is. Copright Big Ideas Learning, LLC
21 Chapter 9. For the equation =, the slope is and the intercept is. Plot (, ). Use the slope to find another point on the line. rise run Plot the point that is unit right and units up from,. Draw a line through the two points. To find the intercept, find the value of when =. = = = = The intercept is.. For the equation = + 9, the slope is and the intercept is 9. Plot (, 9 ). Use the slope to find another point on the line. rise run Plot the point that is units right and units down from (, 9 ). Draw a line through the two points. 9 = = + 9 To find the intercept, find the value of when =. = + 9 = = = The intercept is.. For the equation =., the slope is. and the intercept is. Plot (, ). Use the slope to find another point on the line. rise. run Plot the point that is unit right and. units down,. Draw a line through the two points. from =. To find the intercept, find the value of when =. =. =. =. = = = The intercept is. Copright Big Ideas Learning, LLC
22 Chapter. + 9 = = 9. Plot The slope is, and the intercept is 9., 9. Use the slope to find another point on the line. rise run Plot the point that is unit left and units up from, 9. Draw a line through the two points. + 9 = The line crosses the ais at (, ). So, the intercept is. = = + The slope is, Use the slope to find another point on the line. rise run and the intercept is. Plot (, ). Plot the point that is units right and units down from (, ). Draw a line through the two points. 9 9 =. a. Let be the total cost and be the number of pounds of apples. Equation: = cost per ( ) + admission cost pound An equation is =. +. b. The slope is., and the intercept is. Plot (, ). Use the slope to find another point on the line. rise. = run Plot the point that is units right and units up from (, ). Draw a line through the two points. = a. The equations = + and = are parallel because the slope of each line is. The equations = and = + are parallel because the slope of each line is. b. The equations = + and = + are perpendicular because the product of their slopes is. The equations = and = + are perpendicular because the product of their slopes is. The equations = and = + are perpendicular because the product of their slopes is. To find the intercept, find the value of when =. = = = = The intercept is. Copright Big Ideas Learning, LLC
23 Chapter. a. Let be the monthl income and be the number of clicks. An equation is =.. b. The slope is., and the intercept is. Plot (, ). Use the slope to find another point on the line. rise.. = = run Plot the point that is units right and units up from (, ). Draw a line through the two points. When the ads start to get, clicks per month, the income will be $ per month, which equals the cost of operating the website. Each banner ad needs to average, = clicks. An additional clicks per month will start earning a profit. Fair Game Review. = = +. + = = + = + 9. = = + =. + = = + = +. B; Monthl income 9 = () ( ) =?? (,, ) + = =, clicks,, Number of clicks per month The point (, ) is a solution of the equation. Section.. Activit (pp. ). a. An equation is + =. b. c. 9 Number of Adult Tickets, Number of Student Tickets, The points lie on a line. d. es; If ou remember how man adult tickets ou sold, ou can determine how man student tickets ou sold b substituting the number of adult tickets for in the equation and then solving the equation for.. a. An equation is + =. b. + = = + = + The slope is, and the intercept is. Plot (, ). Use the slope to find another point on the line. rise run Plot the point that is unit right and units down from (, ). Draw a line through the two points = + = = = c. = + 9 You sold pounds of swiss cheese. d. es; You can have. pounds of swiss cheese. Copright Big Ideas Learning, LLC
24 Chapter. The graph of the equation a + b = c is a line. The slope is a, b and the intercept is c b.. In Activit, a table of values is created and the points from the table are plotted. Then a line is drawn through the points. In Activit, the equation is rewritten in slopeintercept form and then the slope and intercept are used to graph the line. Sample answer: The method in Activit ma be considered easier because it involves plotting fewer points.. Sample answer: You bu pounds of dog food for $ a pound and pounds of cat food for $. per pound. Your total bill is $. You can t remember how man pounds of each tpe of food ou bought. $ Pounds of $. Pounds of + = $ lb dog food lb cat food. Graphing the equation using standard form is easier because ou can see that when =, =, and when =, =. You can graph the equation using its intercept and its intercept.. On Your Own (pp. ). + = = The slope is, and the intercept is. Plot (, ). Use the slope to find another point on the line. rise run Plot the point that is unit right and unit down from,. Draw a line through the points. + =.. + = = + = + and the intercept is. Plot (, ). Use The slope is, the slope to find another point on the line. rise run Plot the point that is units right and unit up from (, ). Draw a line through the points. + = + = = and the intercept is. Plot (, ). Use The slope is, the slope to find another point on the line. rise run Plot the point that is units right and units up from (, ). Draw a line through the points. + = Copright Big Ideas Learning, LLC
25 Chapter. + = = + The slope is, and the intercept is. Plot (, ). Use the slope to find another point on the line. rise run Plot the point that is unit right and units down from (, ). Draw a line through the points.. = = = =. The intercept is. = = = = The intercept is. + = + = = The intercept is. + = + = = = + = The intercept is. (, ) + = = (, ) (, ) (, ).. +. =. +.( ) =. = =. +. =.( ) +. =. = = The intercept is. The intercept is. (, ). +. = (, ) The intercept shows that ou can bu pounds of apples when ou do not bu an oranges. The intercept shows that ou can bu pounds of oranges when ou do not bu an apples.. Eercises (pp. ) Vocabular and Concept Check. no; The equation = + is not in standard form because it is not written in the form a + b = c.. Sample answer: ) Write the equation in slopeintercept form and use the slope and intercept to graph the equation. ) Find the  and intercepts, plot the points representing the intercepts, and draw a line through the points. Practice and Problem Solving. Let be the number of pounds of peaches and be the number of pounds of apples. +. =. = + = + The slope is, Use the slope to find another point on the line. rise run Plot the point that is units and the intercept is. Plot (, ). right and units down from (, ). Draw a line through the points. 9 = + 9 Copright Big Ideas Learning, LLC
26 Chapter. Let be the number of hours biked and be the number of hours walked. + = = + = + The slope is, and the intercept is. Plot (, ). Use the slope to find another point on the line. rise run Plot the point that is unit right and units down from (, ). Draw a line through the points.. + = = +.. = + = = + = + = = = 9 = + = + The slope is, and the intercept is. Plot (, ). Use the slope to find another point on the line. rise run Plot the point that is unit right and units up from (, ). Draw a line through the points. + 9 = 9. = = + = The slope is, and the intercept is. Plot,. Use the slope to find another point on the line. rise run Plot the point that is unit right and units up from,. Draw a line through the points. =. + = = + = + The slope is, and the intercept is. Plot,. Use the slope to find another point on the line. rise run Plot the point that is units right and unit down from,. Draw a line through the points.. B; = ( ) = = = + = = ( ) = = = The intercept is. The intercept is. Copright Big Ideas Learning, LLC
27 Chapter. A; + = + ( ) = = = + = + = = = The intercept is. The intercept is.. C; + = + ( ) = = = + = ( ) + = = = The intercept is. The intercept is.. To find the intercept, should have been substituted for, not for. + = + ( ) = = = The intercept is.. a. + = = + The slope is, and the intercept is. Plot (, ). Use the slope to find another point on the line. rise run Plot the point that is unit right and units up from (, ). Draw a line through the points. 9 + = b. The bracelet shown has charms. So, =. = + = + = + = 9 The bracelet shown costs $9... = ( ) = = =. + = + = = = The intercept is. The intercept is. = = = = The intercept is. = (, ) = ( ) = = = The intercept is. = ( ) = = = The intercept is. (, ) = (, ) (, ) + = + = = The intercept is. (, ) + = (, ) Copright Big Ideas Learning, LLC
28 Chapter 9. + = + ( ) = = = 9 The intercept is 9. + = ( ) + = = = The intercept is. Number of DVDs 9 (, ) (9, ) 9 Das Das $ $. a. boat + gear = $ da da rented rented Let be the number of das the boat is rented and let be the number of das the scuba gear is rented. An equation is + =. b. + = + ( ) = = = The intercept is. It shows that the friends could rent the boat for das if the do not rent the scuba gear. + = ( ) + = = = The intercept is. It shows that the friends could rent scuba gear for das if the do not rent the boat. Das gear rented + = Number of CDs Das boat rented Hours Hours $9. $.. a. worked + worked = $. hour hour as host as server Let be the number of hours worked as host and let be the number of hours worked as server. An equation is =.. b = ( ) =. 9. =. = The intercept is =. 9.( ) +. =.. =. = The intercept is.. no; Horizontal lines of the form = a, where a is not equal to, do not have an intercept.. a. Hours worked as server 9 Sample answer: The graph of = does not have an intercept because it does not cross the ais. = =. Hours worked as host O Cost per Number Total fee = Initial fee hour of hours + Let be the total fee charged and let be the number of hours the visit lasts. An equation is = +. Copright Big Ideas Learning, LLC
29 Chapter b. = + = + =. = The intercept is.. This value does not make sense because ou cannot have a negative number of hours. c. From the equation = +, the slope is and the intercept is. Plot (, ). Use the slope to find another point on the line. rise run Plot the point that is unit right and units up from (, ). Draw a line through the points. Fair Game Review ( ). = = The slope is.. Total fee (dollars) 9 = The slope is.. D; = 9 = + = Section. = + Number of hours. Activit (pp. 9) rise. a. Blue line: run Red line: rise run The slope of the red line is. The red line crosses the ais at (, ). So, the intercept is. An equation of the red line is Green line: rise run The slope of the green line is. = +. The green line crosses the ais at (, ). intercept is. An equation of the green line is =. So, the All three lines have a slope of. So, the lines are parallel. rise b. Blue line: = run The slope of the blue line is. The blue line crosses the ais at (, ). So, the intercept is. An equation of the blue line is = +. rise Red line: = run The slope of the red line is. The red line crosses the ais at (, ). So, the intercept is. An equation of the red line is =. rise Green line: = run The slope of the green line is. The green line crosses the ais at (, ). So, the intercept is. An equation of the green line is =. All three lines have a slope of. So, the lines are parallel. The slope of the blue line is. The blue line crosses the ais at (, ). So, the intercept is. An equation of the blue line is Copright Big Ideas Learning, LLC = +. 9
30 Chapter rise c. Blue line: = run The slope of the blue line is. The blue line crosses the ais at (, ). So, the intercept is. An equation of the blue line is Red line: rise run = +. The slope of the red line is. The red line crosses the ais at (, ). So, the intercept is. An equation of the red line is Green line: rise run = +. The slope of the green line is. The green line crosses the ais at (, ). So, the intercept is. An equation of the green line is = +. All three lines have a intercept of. d. Blue line: m = rise = run = The slope of the blue line is. The blue line crosses the ais at (, ). So, the intercept is. An equation of the blue line is =. rise Red line: = run The slope of the red line is. The red line crosses the ais at (, ). So, the intercept is. An equation of the red line is =. rise Green line: run. a. Area = base height = = The area of the parallelogram is square units. The bottom side of the parallelogram is a horizontal line passing through (, ). So, an equation for the bottom side is =. The top side of the parallelogram is a horizontal line passing through (, ). So, an equation for the top side is =. Right side: m = = = = The slope of the right side of the parallelogram is. The side passes through (, ). = m + b = + b = + b = b The intercept is. So, an equation for the right side is = +. Left side: m = = = = The slope of the left side of the parallelogram is. The side passes through (, ). = m + b = ( ) + b = + b = b The intercept is. So, an equation for the left side is =. The slope of the green line is. The green line crosses the ais at (, ). So, the intercept is. An equation of the green line is =. All three lines have a intercept of. Copright Big Ideas Learning, LLC
31 Chapter b. Area = base height = = The area of the parallelogram is square units. The bottom side of the parallelogram is a horizontal line passing through (, ). So, an equation for the bottom side is =. The top side of the parallelogram is a horizontal line passing through (, ). So, an equation for the top side is =. Right side: m = ( ) = = ( ) = The slope of the right side of the parallelogram is. The side passes through (, ). = m + b = ( ) + b = + b = b The intercept is. So, an equation for the right side is = +. ( ) Left side: = = The slope of the left side of the parallelogram is. The side crosses the ais at (, ). So, the intercept is. An equation for the left side is = +.. a. The intercept is. So, at the beginning of the trip, the car was miles from Phoeni. b. = = The slope is. So, the speed of the car was miles per hour. c. The point (, ) represents the end of the trip. So, the trip lasted hours. d. The point (, ) represents the end of the trip. So, at the end of the trip, the car was miles from Phoeni. e. Using the intercept from part (a) and the slope from part (b), an equation is = +.. The slopeintercept form of the equation of a line is = m + b. If ou are given the slope and intercept, substitute the slope for m, and the intercept for b to write an equation of the line. Sample answer: An equation of the line with a slope of and a intercept of is =.. Sample answer: The opposite sides of a parallelogram are parallel. Because one of the sides is represented b the equation = +, the side opposite should have a slope of. Because one of the sides is represented b the equation = +, the side opposite should have a slope of. So, the other two sides can be represented b the equations = + and = +.. On Your Own (pp. ). = =. Because the line crosses the ais at (, ), the intercept is. So, the equation is = +. = = Because the line crosses the ais at (, ), the intercept is. So, the equation is =.. The line is horizontal, so the change in is. change in = change in (, ) (, ) O Because the line crosses the ais at (, ), the intercept is. So, the equation is = +, or =. Copright Big Ideas Learning, LLC
32 Chapter. = = Distance remaining (feet) (, ) (, ) 9 Time (months) Because the line crosses the ais at (, ), the intercept is. So, the equation is = +. The tunnel is complete when the distance remaining is feet. So, find the value of when =. = + = + =. = It takes. months to complete the tunnel.. Eercises (pp. ) Vocabular and Concept Check. Sample answer: Find the ratio of the rise to the run between the intercepts. Practice and Problem Solving. The bottom side of the parallelogram is a horizontal line passing through (, ). So, an equation for the bottom side is =. The top side of the parallelogram is a horizontal line passing through (, ). So, an equation for the top side is =. Right side: m = ( ) = = = The slope of the right side of the parallelogram is. The side passes through (, ). = m + b = + b = + b = b The intercept is. So, an equation for the right side is =. ( ) Left side: = = The slope of the left side of the parallelogram is. The side passes through (, ). So, the intercept is. An equation for the left side is = +.. Sample answer: To write an equation of a line using its graph, find the slope and intercept. Then use the slope and intercept to write an equation of the line in slopeintercept form, = m + b, where m is the slope and b is the intercept. Copright Big Ideas Learning, LLC
33 Chapter. The bottom side of the heagon is a horizontal line passing through (, ). So, an equation for the bottom side is =. The top side of the heagon is a horizontal line passing through (, ). So, an equation for the top side is =. ( ) Bottom right side: = The slope of the bottom right side of the heagon is. The side passes through (, ). = m + b = + b = + b = b The intercept is. So, an equation for the bottom right side is =. Top right side: = = The slope of the top right side of the heagon is The side passes through (, ). = m + b = + b = + b = b. The intercept is. So, an equation for the top right side is = +. Bottom left side: = = The slope of the bottom left side of the heagon is The side passes through (, ). = m + b = ( ) + b = + b = b. The intercept is. So, an equation for the bottom left side is = Top left side: = The slope of the top left side of the heagon is. The side passes through (, ). = m + b = ( ) + b = + b = b The intercept is. So, an equation for the top left side is = +. = = ( ) Because the line crosses the ais at (, ), the intercept is. So, the equation is = +. = = ( ) Because the line crosses the ais at (, ), the intercept is. So, the equation is =. = Because the line crosses the ais at (, ), the intercept is. So, the equation is = = = +. Because the line crosses the ais at (, ), the intercept is. So, the equation is = = +.,, the Because the line crosses the ais at ( ) intercept is. So, the equation is = = =. ( ) Because the line crosses the ais at ( ) intercept is. So, the equation is,, the =. Copright Big Ideas Learning, LLC
34 Chapter. The wrong point was used to find the intercept. The,. So, the intercept is. line crosses the ais at, not. The equation of the line is Growth Number Length Length = + rate of ears at birth =. Let be the length and let be the number of ears. The length at birth is given in inches. Convert inches to feet. in. ft in. = ft The growth rate is given in inches per ear. Convert inches per ear to feet per ear. in. ft r in. An equation is = ft r = +.. The line is horizontal, so the change in is. change in = change in O (, ) (, ) Because the line crosses the ais at (, ), the intercept is. So, the equation is = +, or =.. The line is horizontal, so the change in is. change in = change in (, ) (, ) O Because the line crosses the ais at (, ), the intercept is. So, the equation is = +, or =.. The line is horizontal, so the change in is. change in = change in. Because the line crosses the ais at (, ), the intercept is. So, the equation is = +, or =. Amount Amount Number Flat = + raised per mile of miles donation Let be the amount raised and let be the number of miles. Equation: = m + To find the slope, use the fact that after walking miles, ou have raised $.. = m +. = m() +. = m. = m So, the slope is.. An equation is =. +.. a b. c. O (, ) (, ) Speed (mi/h) (, ) (, ) Braking time (seconds) The point (, ) represents the speed of the automobile before braking. The point (, ) represents the time it takes to come to a complete stop. The line represents the speed of the automobile after seconds of braking. = = Because the line crosses the ais at (, ), the intercept is. So, an equation of the line is = +. Copright Big Ideas Learning, LLC
35 Chapter. Number Rate of Number of pages = use (sheets of + sheets remaining per week) weeks Fair Game Review. Let be the number of pages remaining and let be the number of weeks. After week, ou have % of the sheets left. You started with sheets, so find % of. = m() + (, ) (, ) O (, ). = = m + (, ) Equation: = m + To find the slope m, use the fact that after week, ou have sheets remaining.. C; The line crosses the ais at (, ). So, the intercept is. Section. = m. Activit (pp. ) So, an equation is = +.. a. 9. Sample answer: a. Estimate the heights in the photograph. ft ft ft ft Because the line crosses the ais at (, ), ft Height of earold tree: about ft the intercept is. So, an equation of the line is =. b. Height of earold tree: about ft b. Plot the heights of the two trees. (, ) (, ) O c. The trees are growing at a rate of about feet per ear. Because this would put the height of a earold tree at, it is better to adjust the rate of growth to be about. feet per ear. d. A possible equation for the growth rate is =.. Copright Big Ideas Learning, LLC Because the line crosses the ais at (, ), the intercept is. So, an equation of the line is = +.
36 Chapter c. d.. a c. Because the line crosses the ais at (, ), the intercept is. So, an equation of the line is = +. Because the line crosses the ais at (, ), the intercept is. So, an equation of the line is = +. d. The rise is the change in, or the difference in the coordinates. The run is the change in, or the difference in the coordinates. e. m = f. ( ) m = ( ) m = O O O (, ) Run (, ) Rise This result represents the equation of a line with a,. slope of m that passes through the point ( ). Balance (dollars) A Savings Account 9 t Time (months) The slope is because that is the increase each month. The point (, ) represents our current savings after months. A A = m( t t) A = ( t ) A = t A = t + An equation is A = t +.. a. = m( ) = ( + ) = = = b. = m( ) = = = + ( ) c. = m( ) = = + = + ( ) d. = m( ) = = ( + ) = + The results are the same. The equation in Activit can be used to write equations in slopeintercept form. Copright Big Ideas Learning, LLC
37 Chapter. It is called the pointslope form of the equation of a line because ou use the slope and a point on the line to write the equation of the line. It is important because it is a quick wa to find the equation of a line.. Use the pointslope form of the equation of a line, = m( ). Substitute the slope for m and the,. point for ( ) Sample answer: Write an equation of the line that passes, with slope. through the point = m( ) ( ) = ( ) + = ( ) + = + = + So, the equation is = +.. On Your Own (pp. ). = m( ) = ( ) So, the equation is = (. ). = m( ) = ( ) So, the equation is = ( ).. = m( ). ( ) = ( ) + = ( + ) So, the equation is + = ( + ). = = ( ) = m = = + = = So, the equation is =... = = ( ) = m = = + = + = + So, the equation is = +. 9 = = = m = = ( ) ( + ) = + = + So, the equation is. = m( ) = ( ) = + = + = +. So, the equation is = +. = + (, ). Eercises (pp. 9) Vocabular and Concept Check. The slope is,, is a point on the line. and. a. To write an equation of a line using its slope and a point on the line, write the pointslope form. Substitute the slope for m and the point for (, ). Simplif and check our work. b. To write an equation of a line using two points on the line, first use the two points to find the slope. Then write the pointslope form. Substitute the slope for m and one of the points for (, ). Simplif and check our work. Copright Big Ideas Learning, LLC
38 Chapter Practice and Problem Solving. = m( ) = = ( + ) So, the equation is = ( + ).. = m( ) = = ( + ) So, the equation is = ( + ).. = m( ) ( ) = ( ) + = ( ) So, the equation is + = ( ).. = m( ) = ( ) So, the equation is = ( ).. = m( ) = ( ) So, the equation is = ( ).. = m( ) ( ) = ( ) + = ( ) So, the equation is + = (. ) 9. = m( ) = + = ( ) ( ) ( ) So, the equation is + = ( ).. = m( ) = ( ) So, the equation is = ( ).. = m( ) + = ( + ) = So, the equation is + = ( +. ). = = ( ) = m = = = + So, the equation is = +.. = = ( ) = m = = = So, the equation is =.. = = ( ) = m = = = + So, the equation is = +.. = = m = = = ( ) So, the equation is =. Copright Big Ideas Learning, LLC
39 Chapter.. = = 9 = m = 9 = ( + 9) = = + So, the equation is = +. = = ( ) = m = = = + So, the equation is = +.. For each degree the temperature increases, the volume of the gas increases b. So, the slope is. At C, the volume is liters. So, the line passes through (, ) and the Vintercept is. So, the equation is V = T a. The value decreases $ each ear. So, the slope is. After ears, the value is $,. So, the line passes through (,, ). V V = m( ) V, = ( ) V, = +, V = +, So, the equation is V = +,. b. The original value of the car is represented b the Vintercept. So, the original value was $,.. a. The slope of the line parallel to = is. ( ) = m = = = So, the equation of the parallel line is =. b. The slope of the line perpendicular to = is. ( ) = m = = + = + So, the equation of the perpendicular line is = +.. a. Let be the chirps per minute and let T be the temperature ( F. ) For each rise in temperature of. F, the cricket makes an additional chirp each minute. So, the slope is.. A cricket chirps times per minute when the temperature is F. So, the line passes through (, ). T T = m( ) T =.( ) T =. T =. + So, the equation is T =. +. b. T =. + =.( ) + = + = The temperature is F. c. T =. + 9 =. + =. = You would epect the cricket to make chirps in minute.. a. After seconds, the can contains ounces and after seconds, the can contains ounces. So, the,. line passes through (, ) and = = ( ) = m = = + = + So, the equation is = +. Copright Big Ideas Learning, LLC 9
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