Find the slope of each line.
1.
SOLUTION:
The coordinates of the point J is (–2, 3) and that of K is (3, –2). Substitute the values in the slope formula with (x1,
y1) = (–2, 3) and (x2, y2) = (3, –2).
Therefore, the slope of the line is –1.
ANSWER:
–1
2.
SOLUTION:
Here, isaverticalline.So,theslopeisundefined.
ANSWER:
undefined
3.
SOLUTION:
The coordinates of the point A is andthatofB is . Substitute the values in the slope formula with
(x1, y1) = and (x2, y2) = .
Therefore, the slope of the line is .
ANSWER:
4.BOTANY Kudzu is a fast-growing vine found in the southeastern United States. An initial measurement of the
length of a kudzu vine was 0.5 meter. Seven days later the plant was 4 meters long.
a. Graph the line that models the length of the plant over time.
b. What is the slope of your graph? What does it represent?
c. Assuming that the growth rate of the plant continues, how long will the plant be after 15 days?
SOLUTION:
a.Letx represent the day and y the height. The plant starts at 0.5 m. So one ordered pair is (0, 0.5). After 7 days
the plant grows to 4 m so a second ordered pair is (7, 4) .
Plot the points (0, 0.5) and (7, 4) and join them by a straight line on a coordinate plane.
b. Substitute the values in the slope formula with (x1, y1) =(0, 0.5) and (x2, y2) = (7, 4) .
The slope of the line is
The slope of the line represents the rate of growth of the plant. That is, the plant grows 0.5 m per day.
c. Substitute , x1 = 7, y1 = 4, and x2 = 15 in the slope formula.
Therefore, if the trend continues, the plant will be 8 m long in 15 days.
ANSWER:
a.
b. The plant grows 0.5 m a day.
c. 8 m
Determine whether and areparallel , perpendicular, or neither. Graph each line to verify your
answer.
5.W(2, 4), X(4, 5), Y(4, 1), Z(8, –7)
SOLUTION:
Substitute the coordinates of the points in slope formula to find the slopes of the lines.
Slope
.
Slope
The product of the slopes of the lines is –1. Therefore, the lines are perpendicular.
Graph the lines on a coordinate plane to verify the answer.
ANSWER:
perpendicular
6.W(1, 3), X(–2, –5), Y(–6, –2), Z(8, 3)
SOLUTION:
Substitute the coordinates of the points in slope formula to find the slopes of the lines.
Slope
Slope
The two lines neither have equal slopes nor is their product –1. Therefore, the lines are neither parallel nor
perpendicular.
Graph the lines on a coordinate plane to verify the answer.
ANSWER:
neither
7.W(–7, 6), X(–6, 9), Y(6, 3), Z(3, –6)
SOLUTION:
Substitute the coordinates of the points in slope formula to find the slopes of the lines.
Slope
Slope
The two lines have equal slopes, 3. Therefore, the lines are parallel.
Graph the lines on a coordinate plane to verify the answer.
ANSWER:
parallel
8.W(1, –3), X(0, 2), Y(–2, 0), Z(8, 2)
SOLUTION:
Substitute the coordinates of the points in slope formula to find the slopes of the lines.
Slope
Slope
The product of the slopes of the lines is –1. Therefore, the lines are perpendicular.
Graph the lines on a coordinate plane to verify the answer.
ANSWER:
perpendicular
Graph the line that satisfies each condition.
9.passes through A(3, –4), parallel to withB(2, 4) and C(5, 6)
SOLUTION:
Find the slope of the line with (x1, y1) = (2, 4) and (x2, y2)=(5, 6).
The required line is parallel to . So, the slope of the required line is also
Start at A(3, –4). Move two units up and three units right to reach the point (6, –2). Join the two points and extend.
ANSWER:
10.slope = 3, passes through A(–1, 4)
SOLUTION:
Start at A(–1, 4). Move three units up and one unit right to reach the point (0, 7). Join the two points and extend.
ANSWER:
11.passes through P(7, 3), perpendicular to withL(–2, –3) and M(–1, 5)
SOLUTION:
Find the slope of the line with and (x1, y1) = (–2, –3) and (x2, y2) = (–1, 5).
The required line is perpendicular to . So, the slope of the required line is
Start at P(7, 3). Move one unit up and eight units left to reach the point (–1, 4). Join the two points and extend.
ANSWER:
Find the slope of each line.
12.
SOLUTION:
The coordinates of the point C is (–2, –1) and that of D is (5, 5). Substitute the values in the slope formula.
Therefore, the slope of the line is
ANSWER:
13.
SOLUTION:
The coordinates of the point A is (–3, 1) and that of B is (4, –2). Substitute the values in the slope formula.
Therefore, the slope of the line is
ANSWER:
14.
SOLUTION:
Here, isahorizontalline.So,theslopeiszero.
ANSWER:
0
15.
SOLUTION:
The coordinates of the point X is (2, 4) and that of Y is (1, –4). Substitute the values in the slope formula.
Therefore, the slope of the line is 8.
ANSWER:
8
16.
SOLUTION:
The coordinates of the point M is (–4, 1) and that of N is (1, –3). Substitute the values in the slope formula.
Therefore, the slope of the line is
ANSWER:
17.
SOLUTION:
Here, isaverticalline.So,theslopeisundefined.
ANSWER:
undefined
Determine the slope of the line that contains the given points.
18.C(3, 1), D(–2, 1)
SOLUTION:
Substitute the coordinates of the points in the slope formula.
Therefore, the slope of the line is 0.
ANSWER:
0
19.E(5, –1), F(2, –4)
SOLUTION:
Substitute the coordinates of the points in the slope formula.
Therefore, the slope of the line is 1.
ANSWER:
1
20.G(–4, 3), H(–4, 7)
SOLUTION:
Substitute the coordinates of the points in the slope formula.
Division of any number by zero is undefined.
Therefore, the slope of the line is undefined.
ANSWER:
undefined
21.J(7, –3), K(–8, –3)
SOLUTION:
Substitute the coordinates of the points in the slope formula.
Therefore, the slope of the line is 0.
ANSWER:
0
22.L(8, –3), M(–4, –12)
SOLUTION:
Substitute the coordinates of the points in the slope formula.
Therefore, the slope of the line is
ANSWER:
23.P(–3, –5), Q(–3, –1)
SOLUTION:
Substitute the coordinates of the points in the slope formula.
Division of any number by zero is undefined.
Therefore, the slope of the line is undefined.
ANSWER:
undefined
24.R(2, –6), S(–6, 5)
SOLUTION:
Substitute the coordinates of the points in the slope formula.
Therefore, the slope of the line is
ANSWER:
25.T(–6, –11), V(–12, –10)
SOLUTION:
Substitute the coordinates of the points in the slope formula.
Therefore, the slope of the line is
ANSWER:
26.CCSS MODELING In 2004, 8 million Americans over the age of 7 participated in mountain biking, and in 2006, 8.5
million participated.
a. Create a graph to show the number of participants in mountain biking based on the change in participation from
2004 to 2006.
b. Based on the data, what is the growth per year of the sport?
c. If participation continues at the same rate, what will be the participation in 2013 to the nearest 10,000?
SOLUTION:
a. Plot the points (2004, 8) and (2006, 8.5), join them and extend the line.
b. Substitute the coordinates of any two points on the line in the slope formula. Consider the points (2004, 8) and
(2012, 10) since neither has a decimal and will make the slope calculation easier.
Let (x1, y1)= (2004, 8) and (x2, y2)= (2012, 10). Find m.
The rate of growth is or0.25.Thatis,250,000peopleperyear.
c. Substitute , x1 = 2004, y1 = 8, and x2 = 2013 in the slope formula.
Therefore, if the trend continues the participation in 2013 will be about 10.25 million.
ANSWER:
a.
b. 250,000 people per year
c. 10,250,000
27.FINANCIAL LITERACY Suppose an MP3 player cost $499 in 2003 and $249.99 in 2009.
a. Graph a trend line to predict the price of the MP3 player for 2003 through 2009.
b. Based on the data, how much does the price drop per year?
c. If the trend continues, what will be the cost of an MP3 player in 2013?
SOLUTION:
a. Plot the points (2003, 499) and (2009, 249.99), join them and extend the line.
b. Substitute the coordinates of any two points on the line in the slope formula. Consider the points (2003, 499) and
(2009, 249.99)
The price drops $41.50 per year.
c. Substitute m = –41.50, x1 = 2003, y1 = 499, and x2 = 2013 in the slope formula.
Therefore, if the trend continues an MP3 Player will cost $84 in 2013.
ANSWER:
a.
b. $41.50
c. $84
Determine whether and areparallel, perpendicular, or neither. Graph each line to verify your
answer.
28.A(1, 5), B(4, 4), C(9, –10), D(–6, –5)
SOLUTION:
Substitute the coordinates of the points in slope formula to find the slopes of the lines.
Find slope of with (x1, y1) = (1, 5) and (x2, y2) =(4,4).
Find slope of with (x1, y1)=(9, –10) and (x2, y2)= (–6, –5).
The two lines have equal slopes, Therefore,thelinesareparallel.
Graph the lines on a coordinate plane to verify the answer.
ANSWER:
parallel
29.A(–6, –9), B(8, 19), C(0, –4), D(2, 0)
SOLUTION:
Substitute the coordinates of the points in slope formula to find the slopes of the lines.
Find slope of with (x1, y1) = (–6, –9) and (x2, y2) =(8,19).
Find slope of with (x1, y1) = (0, –4) and (x2, y2) =(2,0).
The two lines have equal slopes, 2. Therefore, the lines are parallel.
Graph the lines on a coordinate plane to verify the answer.
ANSWER:
parallel
30.A(4, 2), B(–3, 1), C(6, 0), D(–10, 8)
SOLUTION:
Substitute the coordinates of the points in slope formula to find the slopes of the lines.
Find slope of with (x1, y1) = (4, 2) and (x2, y2) = (–3,1).
Find slope of with (x1, y1) = (6, 0) and (x2, y2) = (–10,8).
The two lines neither have equal slopes nor is their product –1. Therefore, the lines are neither parallel nor
perpendicular.
Graph the lines on a coordinate plane to verify the answer.
ANSWER:
neither
31.A(8, –2), B(4, –1), C(3, 11), D(–2, –9)
SOLUTION:
Substitute the coordinates of the points in slope formula to find the slopes of the lines.
Find slope of with (x1, y1) = (8, –2) and (x2, y2)=(4,–1).
Find slope of with (x1, y1) =C(3, 11) and (x2, y2) = D(–2, –9).
The product of the slopes of the lines is –1. Therefore, the lines are perpendicular.
Graph the lines on a coordinate plane to verify the answer.
ANSWER:
perpendicular
32.A(8, 4), B(4, 3), C(4, –9), D(2, –1)
SOLUTION:
Substitute the coordinates of the points in slope formula to find the slopes of the lines.
Find slope of with (x1, y1) = (8, 4) and (x2, y2) =(4,3).
Find slope of with (x1, y1) = (4, –9) and (x2, y2) = (2, –1).
The product of the slopes of the lines is –1. Therefore, the lines are perpendicular.
Graph the lines on a coordinate plane to verify the answer.
ANSWER:
perpendicular
33.A(4, –2), B(–2, –8), C(4, 6), D(8, 5)
SOLUTION:
Substitute the coordinates of the points in slope formula to find the slopes of the lines.
Find slope of with (x1, y1) = (4, –2) and (x2, y2) = (–2, –8).
Find slope of with (x1, y1) = (4, 6) and (x2, y2) =(8,5).
The two lines neither have equal slopes nor is their product –1. Therefore, the lines are neither parallel nor
perpendicular.
Graph the lines on a coordinate plane to verify the answer.
ANSWER:
neither
Graph the line that satisfies each condition.
34.passes through A(2, –5), parallel to withB(1, 3) and C(4, 5)
SOLUTION:
Find the slope of the line with (x1, y1) =(1, 3) and (x2, y2) =(4,5).
The required line is parallel to . So, the slope of the required line is also .
Start at A(2, –5). Move two units up and three unit right to reach the point (5, –3). Join the two points and extend.
ANSWER:
35.slope = –2, passes through H(–2, –4)
SOLUTION:
Start at H(–2, –4). Move two units up and one unit left to reach the point (–3, –2). Join the two points and extend.
ANSWER:
36.passes through K(3, 7), perpendicular to withL(–1, –2) and M(–4, 8)
SOLUTION:
Find the slope of the line with (x1, y1) = (–1, –2) and (x2, y2) = (–4, 8).
The required line is perpendicular to . So, the slope of the required line is
Start at K(3, 7). Move three units up and ten units right to reach the point (13, 10). Join the two points and extend.
ANSWER:
37.passes through X(1, –4), parallel to withY(5, 2) and Z(–3, –5)
SOLUTION:
Find the slope of the line with (x1, y1) = (5, 2) and (x2, y2) = (–3, –5).
The required line is parallel to . So, the slope of the required line is also .
Start at X(1, –4). Move seven units up and eight units right to reach the point (9, 3). Join the two points and extend.
ANSWER:
38.slope = , passes through J(–5, 4)
SOLUTION:
Start at J(–5, 4). Move two units up and three units right to reach the point (–2, 6). Join the two points and extend.
ANSWER:
39.passes through D(–5, –6), perpendicular to withF(–2, –9) and G(1, –5)
SOLUTION:
Find the slope of the line with (x1, y1) = (–2, –9) and (x2, y2) = (1, –5).
The required line is perpendicular to . So, the slope of the required line is
Start at D(–5, –6). Move three units down and four units to right to reach the point (–1, –9). Join the two points and
extend.
ANSWER:
40.STADIUMS Before it was demolished, the RCA Dome was home to the Indianapolis Colts. The attendance in
2001 was 450,746, and the attendance in 2005 was 457,373.
a. What is the approximate rate of change in attendance from 2001 to 2005?
b. If this rate of change continues, predict the attendance for 2012.
c. Will the attendance continue to increase indefinitely? Explain.
d. The Colts have now built a new, larger stadium. Do you think their decision was reasonable? Why or why not?
SOLUTION:
a. Substitute the coordinates of any two points on the line in the slope formula. Consider the points (2001, 450,746)
and (2005, 457,373). Let (x1, y1)=(2001, 450,746) and (x2, y2) = (2005, 457,373).
The rate of growth is about 1657. That is, the attendance increases about 1657 per year.
b. To find the attendance in 2012, let m = 1657, x1 = 2001, y1 = 450746, and x2 = 2012. Then use the slope formula
to find y2.
Therefore, if the rate of change continues the attendance will be about 468,973 in 2012.
c. No; the attendance could only continue to increase until the capacity of the stadium was reached.
d. Sample answer: Yes; since their attendance is growing, the new stadium will allow them to accommodate more
fans.
ANSWER:
a. 1657
b. 468,973
c. No; the attendance could only continue to increase until the capacity of the stadium was reached.
d. Sample answer: Yes; since their attendance is growing, the new stadium will allow them to accommodate more
fans.
Determine which line passing through the given points has a steeper slope.
41.Line 1: (0, 5) and (6, 1)
Line 2: (–4, 10) and (8, –5)
SOLUTION:
The greater the absolute value of the slope, the steeper the slope of the line becomes.
Substitute the coordinates of the points in slope formula to find the slopes of the two lines and then compare their
absolute values to determine which is steeper.
Since , Therefore,line2issteeperthanline1.
ANSWER:
Line 2
42.Line 1: (0, –4) and (2, 2)
Line 2: (0, –4) and (4, 5)
SOLUTION:
The greater the absolute value of the slope, the steeper the slope of the line becomes.
Substitute the coordinates of the points in slope formula to find the slopes of the two lines and then compare their
absolute values to determine which is steeper.
Since , Therefore,line1issteeperthanline2.
ANSWER:
Line 1
43.Line 1: (–9, –4) and (7, 0)
Line 2: (0, 1) and (7, 4)
SOLUTION:
The greater the absolute value of the slope, the steeper the slope of the line becomes.
Substitute the coordinates of the points in slope formula to find the slopes of the two lines and then compare their
absolute values to determine which is steeper.
Since , Therefore,line2issteeperthanline1.
ANSWER:
Line 2
44.Line 1: (–6, 7) and (9, –3)
Line 2: (–9, 9) and (3, 5)
SOLUTION:
The greater the absolute value of the slope, the steeper the slope of the line becomes.
Substitute the coordinates of the points in slope formula to find the slopes of the two lines and then compare their
absolute values to determine which is steeper.
Since , Therefore,line1issteeperthanline2.
ANSWER:
Line 1
45.CCSS MODELING Michigan provides habitat for two endangered species, the bald eagle and the gray wolf. The
graph shows the Michigan population of each species in 1992 and 2006.
a. Which species experienced a greater rate of change in population?
b. Make a line graph showing the growth of both populations.
c. If both species continue to grow at their respective rates, what will the population of each species be in 2012?
SOLUTION:
a. Use the points (1992, 440), (2006, 964), (1992, 50), and (2006, 361) to find the rate of change (slope) for
eachspeciesoverthegiventimeperiods.
The bald eagle increased at a rate of about 37.43 per year and the gray wolf increased at a rate of about 22.21 per
year. So, the bald eagle had a greater rate of change in population.
b. Plot the points (1992, 440) and (2006, 964), join them and extend to get the line representing the growth of the bald
eagle.
Similarly, plot the points (1992, 50) and (2006, 361), join them and extend to get the line representing the growth of
the bald eagle.
c.Substitute m = 37.43, x1 = 2006, y1 = 964, and x2 = 2012 in the slope formula to find the population of bald eagle
in 2012.
Substitute m = 22.21, x1 = 1992, y1 = 50, and x2 = 2012 in the slope formula to find the population of gray wolf in
2012.
Therefore, if the rate of change continues there will be about 1189 bald eagles and 494 gray wolves in 2012.
ANSWER:
a. the bald eagle
b.
c. 1189 bald eagles; 494 gray wolves
Find the value of x or y that satisfies the given conditions. Then graph the line.
46.The line containing (4, –1) and (x, –6) has a slope of
SOLUTION:
Substitute the coordinates of the points and the value of slope in the slope formula.
Let(x1, y1) = (4, –10, (x2, y2) =(x, –6) and m = .
Plot the points (4, –1) and (6, –6) and join them by a straight line.
ANSWER:
;
47.The line containing (–4, 9) and (4, 3) is parallel to the line containing (–8, 1) and (4, y).
SOLUTION:
Find the slope of the line containing (–4,9)and(4,3)with(x1, y1) =(–4, 9)and (x2, y2) =(4, 3).
The two lines are parallel, hence they have the same slope.
Substitute the coordinates of the points and the value of slope in the slope formula. Let (x1, y1) = (–8,1),(x2, y2) =
(4, y), and m = .
Plot the points (–8, 1) and (4, –8) and join them by a straight line.
ANSWER:
48.The line containing (8, 7) and (7, –6) is perpendicular to the line containing (2, 4) and (x, 3).
SOLUTION:
Find the slope of the line containing (8, 7) and (7, –6) with (x1, y1) = (8, 7) and (x2, y2) =(7, –6).
The two lines are perpendicular; hence the product of their slopes is –1.
So, the slope of the line passing through (2, 4) and (x, 3) is .
Substitute the coordinates of the points and the value of slope in the slope formula. Let (x1, y1) = (2, 4), (x2, y2) =
(x, 3) and m = .
Plot the points (2, 4) and (15, 3) and join them by a straight line.
ANSWER:
;
49.The line containing (1, –3) and (3, y) is parallel to the line containing (5, –6) and (9, y).
SOLUTION:
The two lines are parallel, hence have the same slope.
Substitutethecoordinatesofthepointsintheslopeformula.
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3-3 Slopes of Lines
