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College Algebra 7th

James Stewart, Lothar Redlin, Saleem Watson

Chapter 5

Systems of Equations and Inequalities

Educators

AG

Problem 1

The system of equations $$\left\{\begin{array}{l}
2 x+3 y=7 \\
5 x-y=9
\end{array}\right.$$
is a system of two equations in the two variables ____ and ______. To determine whether (5,-1) is a solution of this system, we check whether $x=5$ and $y=-1$ satisfy each _____ in the system. Which of the following are solutions of this system?
$$(5,-1), \quad(-1,3), \quad(2,1)$$

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Problem 2

A system of equations in two variables can be solved by the _____ method, the ______ method, or the ______ method.

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Problem 3

A system of two linear equations in two variables can have one solution, ______ solution, or ______ solutions.

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Problem 4

The following is a system of two linear equations in two variables. $$\left\{\begin{aligned}
x+y &=1 \\
2 x+2 y &=2
\end{aligned}\right.$$
The graph of the first equation is the same as the graph of the second equation, so the system has ___ ____ solutions. We express these solutions by writing $$x=t$$ $$y=$$_______
where $t$ is any real number. Some of the solutions of this system are (1,_____) (-3, ____), and (5, _____)_.

AG
Ankit G.
Numerade Educator

Problem 5

Use the substitution method to find all solutions of the system of equations.
$$\left\{\begin{aligned}
x-y &=1 \\
4 x+3 y &=18
\end{aligned}\right.$$

AG
Ankit G.
Numerade Educator

Problem 6

Use the substitution method to find all solutions of the system of equations.
$$\left\{\begin{array}{l}
3 x+y=1 \\
5 x+2 y=1
\end{array}\right.$$

AG
Ankit G.
Numerade Educator

Problem 7

Use the substitution method to find all solutions of the system of equations.
$$\left\{\begin{aligned}
x-y &=2 \\
2 x+3 y &=9
\end{aligned}\right.$$

AG
Ankit G.
Numerade Educator

Problem 8

Use the substitution method to find all solutions of the system of equations.
$$\left\{\begin{aligned}
2 x+y &=7 \\
x+2 y &=2
\end{aligned}\right.$$

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Problem 9

Use the elimination method to find all solutions of the system of equations.
$$\left\{\begin{aligned}
3 x+4 y &=10 \\
x-4 y &=-2
\end{aligned}\right.$$

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Problem 10

Use the elimination method to find all solutions of the system of equations.
$$\left\{\begin{array}{l}
2 x+5 y=15 \\
4 x+y=21
\end{array}\right.$$

AG
Ankit G.
Numerade Educator

Problem 11

Use the elimination method to find all solutions of the system of equations.
$$\left\{\begin{aligned}
3 x-2 y &=-13 \\
-6 x+5 y &=28
\end{aligned}\right.$$

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Problem 12

Use the elimination method to find all solutions of the system of equations.
$$\left\{\begin{array}{l}
2 x-5 y=-18 \\
3 x+4 y=19
\end{array}\right.$$

AG
Ankit G.
Numerade Educator

Problem 13

Two equations and their graphs are given. Find the intersection point(s) of the graphs by solving the system.
$$\left\{\begin{aligned}
2 x+y &=-1 \\
x-2 y &=-8
\end{aligned}\right.$$

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Problem 14

Two equations and their graphs are given. Find the intersection point(s) of the graphs by solving the system.
$$\left\{\begin{aligned}
x+y &=2 \\
2 x+y &=5
\end{aligned}\right.$$

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Problem 15

Graph each linear system, either by hand or using a graphing device. Use the graph to determine whether the system has one solution, no solution, or infinitely many solutions. If there is exactly one solution, use the graph to find it.
$$\left\{\begin{aligned}
x-y &=4 \\
2 x+y &=2
\end{aligned}\right.$$

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Problem 16

Graph each linear system, either by hand or using a graphing device. Use the graph to determine whether the system has one solution, no solution, or infinitely many solutions. If there is exactly one solution, use the graph to find it.
$$\left\{\begin{array}{l}
2 x-y=4 \\
3 x+y=6
\end{array}\right.$$

AG
Ankit G.
Numerade Educator

Problem 17

Graph each linear system, either by hand or using a graphing device. Use the graph to determine whether the system has one solution, no solution, or infinitely many solutions. If there is exactly one solution, use the graph to find it.
$$\left\{\begin{array}{ccc}
2 x-3 y & = & 12 \\
-x+ & \frac{3}{2} y & = & 4
\end{array}\right.$$

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Problem 18

Graph each linear system, either by hand or using a graphing device. Use the graph to determine whether the system has one solution, no solution, or infinitely many solutions. If there is exactly one solution, use the graph to find it.
$$\left\{\begin{aligned}
2 x+6 y &=0 \\
-3 x-9 y &=18
\end{aligned}\right.$$

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Problem 19

Graph each linear system, either by hand or using a graphing device. Use the graph to determine whether the system has one solution, no solution, or infinitely many solutions. If there is exactly one solution, use the graph to find it.
$$\left\{\begin{array}{l}
-x+\frac{1}{2} y=-5 \\
2 x-y=10
\end{array}\right.$$

AG
Ankit G.
Numerade Educator

Problem 20

Graph each linear system, either by hand or using a graphing device. Use the graph to determine whether the system has one solution, no solution, or infinitely many solutions. If there is exactly one solution, use the graph to find it.
$$\left\{\begin{array}{l}
2 x+15 y=-18 \\
2 x+\frac{5}{2} y=-3
\end{array}\right.$$

AG
Ankit G.
Numerade Educator

Problem 21

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 6
$$\left\{\begin{aligned}
x+y &=4 \\
-x+y &=0
\end{aligned}\right.$$

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Problem 22

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 6
$$\left\{\begin{array}{l}
x-y=3 \\
x+3 y=7
\end{array}\right.$$

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Problem 23

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 6
$$\left\{\begin{array}{l}
2 x-3 y=9 \\
4 x+3 y=9
\end{array}\right.$$

AG
Ankit G.
Numerade Educator

Problem 24

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 6
$$\left\{\begin{aligned}
3 x+2 y &=0 \\
-x-2 y &=8
\end{aligned}\right.$$

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Problem 25

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 6
$$\left\{\begin{aligned}
x+3 y &=5 \\
2 x-y &=3
\end{aligned}\right.$$

AG
Ankit G.
Numerade Educator

Problem 26

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 6
$$\left\{\begin{aligned}
x+y &=7 \\
2 x-3 y &=-1
\end{aligned}\right.$$

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Problem 27

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 6
$$\left\{\begin{array}{rr}
-x+y= & 2 \\
4 x-3 y= & -3
\end{array}\right.$$

AG
Ankit G.
Numerade Educator

Problem 28

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 6
$$\left\{\begin{array}{l}
4 x-3 y=28 \\
9 x-y=-6
\end{array}\right.$$

AG
Ankit G.
Numerade Educator

Problem 29

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 6
$$\left\{\begin{aligned}
x+2 y &=7 \\
5 x-y &=2
\end{aligned}\right.$$

AG
Ankit G.
Numerade Educator

Problem 30

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 6
$$\left\{\begin{array}{l}
-4 x+12 y=0 \\
12 x+4 y=160
\end{array}\right.$$

AG
Ankit G.
Numerade Educator

Problem 31

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 6
$$\left\{\begin{aligned}
-\frac{1}{3} x-\frac{1}{6} y &=-1 \\
\frac{2}{3} x+\frac{1}{6} y &=3
\end{aligned}\right.$$

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Problem 32

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 6
$$\left\{\begin{array}{c}
\frac{3}{4} x+\frac{1}{2} y=5 \\
-\frac{1}{4} x-\frac{3}{2} y=1
\end{array}\right.$$

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Problem 33

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 6
$$\left\{\begin{array}{l}
\frac{1}{2} x+\frac{1}{3} y=2 \\
\frac{1}{5} x-\frac{2}{3} y=8
\end{array}\right.$$

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Problem 34

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 6
$$\left\{\begin{aligned}
0.2 x-0.2 y &=-1.8 \\
-0.3 x+0.5 y &=3.3
\end{aligned}\right.$$

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Problem 35

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 6
$$\left\{\begin{aligned}
3 x+2 y &=8 \\
x-2 y &=0
\end{aligned}\right.$$

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Problem 36

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 6
$$\left\{\begin{aligned}
4 x+2 y &=16 \\
x-5 y &=70
\end{aligned}\right.$$

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Problem 37

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 6
$$\left\{\begin{aligned}
x+4 y &=8 \\
3 x+12 y &=2
\end{aligned}\right.$$

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Problem 38

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 6
$$\left\{\begin{aligned}
-3 x+5 y &=2 \\
9 x-15 y &=6
\end{aligned}\right.$$

AG
Ankit G.
Numerade Educator

Problem 39

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 6
$$\left\{\begin{aligned}
2 x-6 y &=10 \\
-3 x+9 y &=-15
\end{aligned}\right.$$

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Problem 40

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 6
$$\left\{\begin{aligned}
2 x-3 y &=-8 \\
14 x-21 y &=3
\end{aligned}\right.$$

AG
Ankit G.
Numerade Educator

Problem 41

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 6
$$\left\{\begin{array}{l}
6 x+4 y=12 \\
9 x+6 y=18
\end{array}\right.$$

AG
Ankit G.
Numerade Educator

Problem 42

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 6
$$\left\{\begin{aligned}
25 x-75 y &=100 \\
-10 x+30 y &=-40
\end{aligned}\right.$$

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Problem 43

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 6
$$\left\{\begin{array}{l}
8 s-3 t=-3 \\
5 s-2 t=-1
\end{array}\right.$$

AG
Ankit G.
Numerade Educator

Problem 44

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 6
$$\left\{\begin{aligned}
u-30 v &=-5 \\
-3 u+80 v &=5
\end{aligned}\right.$$

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Problem 45

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 6
$$\left\{\begin{array}{l}
\frac{1}{2} x+\frac{3}{5} y=3 \\
\frac{5}{3} x+2 y=10
\end{array}\right.$$

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Problem 46

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 6
$$\left\{\begin{aligned}
\frac{3}{2} x-\frac{1}{3} y &=\frac{1}{2} \\
2 x-\frac{1}{2} y &=-\frac{1}{2}
\end{aligned}\right.$$

AG
Ankit G.
Numerade Educator

Problem 47

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 6
$$\left\{\begin{array}{l}
0.4 x+1.2 y=14 \\
12 x-5 y=10
\end{array}\right.$$

AG
Ankit G.
Numerade Educator

Problem 48

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 6
$$\left\{\begin{aligned}
26 x-10 y &=-4 \\
-0.6 x+1.2 y &=3
\end{aligned}\right.$$

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Problem 49

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 6
$$\left\{\begin{array}{ccccc}
\frac{1}{3} x & - & \frac{1}{4} y & = & 2 \\
-8 x & + & 6 y & = & 10
\end{array}\right.$$

AG
Ankit G.
Numerade Educator

Problem 50

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 6
$$\left\{\begin{aligned}
-\frac{1}{10} x+\frac{1}{2} y &=4 \\
2 x-10 y &=-80
\end{aligned}\right.$$

AG
Ankit G.
Numerade Educator

Problem 51

Use a graphing device to graph both lines in the same viewing rectangle. (Note that you must solve for $y$ in terms of $x$ before graphing if you are using a graphing calculator.) Solve the system either by zooming in and using $[\mathrm{TRACE}]$ or by using Intersect. Round your answers to two decimals.
$$\left\{\begin{array}{l}
0.21 x+3.17 y=9.51 \\
2.35 x-1.17 y=5.89
\end{array}\right.$$

AG
Ankit G.
Numerade Educator

Problem 52

Use a graphing device to graph both lines in the same viewing rectangle. (Note that you must solve for $y$ in terms of $x$ before graphing if you are using a graphing calculator.) Solve the system either by zooming in and using $[\mathrm{TRACE}]$ or by using Intersect. Round your answers to two decimals.
$$\left\{\begin{array}{l}
18.72 x-14.91 y=12.33 \\
6.21 x-12.92 y=17.82
\end{array}\right.$$

AG
Ankit G.
Numerade Educator

Problem 53

Use a graphing device to graph both lines in the same viewing rectangle. (Note that you must solve for $y$ in terms of $x$ before graphing if you are using a graphing calculator.) Solve the system either by zooming in and using $[\mathrm{TRACE}]$ or by using Intersect. Round your answers to two decimals.
$$\left\{\begin{array}{l}
2371 x-6552 y=13,591 \\
9815 x+\quad 992 y=618,555
\end{array}\right.$$

AG
Ankit G.
Numerade Educator

Problem 54

Use a graphing device to graph both lines in the same viewing rectangle. (Note that you must solve for $y$ in terms of $x$ before graphing if you are using a graphing calculator.) Solve the system either by zooming in and using $[\mathrm{TRACE}]$ or by using Intersect. Round your answers to two decimals.
$$\left\{\begin{aligned}
-435 x+912 y &=0 \\
132 x+455 y &=994
\end{aligned}\right.$$

AG
Ankit G.
Numerade Educator

Problem 55

Find $x$ and $y$ in terms of $a$ and $b$
$$\left\{\begin{array}{ll}
x+y=0 \\
x+a y=1
\end{array} \quad(a \neq 1)\right.$$

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Problem 56

Find $x$ and $y$ in terms of $a$ and $b$
$$\left\{\begin{aligned}
a x+b y &=0 \\
x+y &=1
\end{aligned} \quad(a \neq b)\right.$$

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Problem 57

Find $x$ and $y$ in terms of $a$ and $b$
$$\left\{\begin{array}{l}
a x+b y=1 \\
b x+a y=1
\end{array} \quad\left(a^{2}-b^{2} \neq 0\right)\right.$$

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Problem 58

Find $x$ and $y$ in terms of $a$ and $b$
$$\left\{\begin{aligned}
a x+b y &=0 \\
a^{2} x+b^{2} y &=1
\end{aligned} \quad(a \neq 0, b \neq 0, a \neq b)\right.$$

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Problem 59

Find two numbers whose sum is 34 and whose difference is 10 .

AG
Ankit G.
Numerade Educator

Problem 60

Number Problem The sum of two numbers is twice their difference. The larger number is 6 more than twice the smaller. Find the numbers.

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Problem 61

Value of Coins A man has 14 coins in his pocket, all of which are dimes and quarters. If the total value of his change is $\$ 2.75$, how many dimes and how many quarters does he have?

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Problem 62

The admission fee at an amusement park is $\$ 1.50$ for children and $\$ 4.00$ for adults. On a certain day, 2200 people entered the park, and the admission fees that were collected totaled $\$ 5050 .$ How many children and how many adults were admitted?

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Problem 63

Gas Station A gas station sells regular gas for $\$ 2.20$ per gallon and premium gas for $\$ 3.00$ a gallon. At the end of a business day 280 gallons of gas had been sold, and receipts totaled $\$ 680 .$ How many gallons of each type of gas had been sold?

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Problem 64

Fruit Stand A fruit stand sells two varieties of strawberries: standard and deluxe. A box of standard strawberries sells for $\$ 7,$ and a box of deluxe strawberries sells for $\$ 10 .$ In one day the stand sold 135 boxes of strawberries for a total of $\$ 1110 .$ How many boxes of each type were sold?

AG
Ankit G.
Numerade Educator

Problem 65

A man flies a small airplane from Fargo to Bismarck, North Dakota-a distance of 180 mi. Because he is flying into a headwind, the trip takes him 2 h. On the way back, the wind is still blowing at the same speed, so the return trip takes only $1 \mathrm{h}$ 12 min. What is his speed in still air, and how fast is the wind blowing?

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Problem 66

A boat on a river travels downstream between two points, 20 mi apart, in 1 h. The return trip against the current takes $2 \frac{1}{2}$ h. What is the boat's speed, and how fast does the current
in the river flow?

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Problem 67

A researcher performs an experiment to test a hypothesis that involves the nutrients niacin and retinol. She feeds one group of laboratory rats a daily diet of precisely 32 units of niacin and 22,000 units of retinol. She uses two types of commercial pellet foods. Food A contains 0.12 unit of niacin and 100 units of retinol per gram. Food $B$ contains 0.20 unit of niacin and 50 units of retinol per gram. How many grams of each food does she feed this group of rats each day?

AG
Ankit G.
Numerade Educator

Problem 68

Coffee Blends A customer in a coffee shop purchases a blend of two coffees: Kenyan, costing $3.50 a pound, and Sri Lankan, costing $\$ 5.60$ a pound. He buys 3 lb of the blend, which costs him \$11.55. How many pounds of each kind went into the mixture?

AG
Ankit G.
Numerade Educator

Problem 69

Mixture Problem A chemist has two large containers of sulfuric acid solution, with different concentrations of acid in each container. Blending 300 mL of the first solution and 600 mL of the second gives a mixture that is $15 \%$ acid, whereas blending 100 mL of the first with 500 mL of the second gives a $12 \frac{1}{2} \%$ acid mixture. What are the concentrations of sulfuric acid in the original containers?

AG
Ankit G.
Numerade Educator

Problem 70

Mixture Problem A biologist has two brine solutions, one containing $5 \%$ salt and another containing $20 \%$ salt. How many milliliters of each solution should she mix to obtain 1 L of a solution that contains 14\% salt?

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Problem 71

Investments A woman invests a total of $\$ 20,000$ in two accounts, one paying $5 \%$ and the other paying $8 \%$ simple interest per year. Her annual interest is $\$ 1180 .$ How much did she invest at each rate?

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Problem 72

Investments A man invests his savings in two accounts, one paying $6 \%$ and the other paying $10 \%$ simple interest per year. He puts twice as much in the lower-yielding account because it is less risky. His annual interest is $\$ 3520 .$ How much did he invest at each rate?

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Problem 73

Distance, Speed, and Time John and Mary leave their house at the same time and drive in opposite directions. John drives at $60 \mathrm{mi} / \mathrm{h}$ and travels $35 \mathrm{mi}$ farther than Mary, who drives at
40 mi/h. Mary's trip takes 15 min longer than John's. For what length of time does each of them drive?

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Problem 74

Aerobic Exercise A woman keeps fit by bicycling and running every day. On Monday she spends $\frac{1}{2} \mathrm{h}$ at each activity, covering a total of $12 \frac{1}{2} \mathrm{mi}$. On Tuesday she runs for 12 min and cycles for 45 min, covering a total of 16 mi. Assuming that her running and cycling speeds don't change from day to day, find these speeds.

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Problem 75

Number Problem The sum of the digits of a two-digit number is $7 .$ When the digits are reversed, the number is increased by 27. Find the number.

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Problem 76

Find the area of the triangle that lies in the first quadrant (with its base on the $x$ -axis) and that is bounded by the lines $y=2 x-4$ and $y=-4 x+20$

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Problem 77

Discuss: The Least Squares Line The least squares line or regression line is the line that best fits a set of points in the plane. We studied this line in the Focus on Modeling that follows Chapter 1. By using calculus, it can be shown that the line that best fits the $n$ data points $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)$ is the line $y=a x+b,$ where the coefficients $a$ and $b$ satisfy the following pair of linear equations. (The notation $\sum_{k=1}^{n} x_{k}$ stands for the sum of all the $x$ 's. See Section 8.1 for a complete description of sigma ( $\Sigma$ ) notation.)
$$\left(\sum_{k=1}^{n} x_{k}\right) a+n b=\sum_{k=1}^{n} y_{k}$$
$$\left(\sum_{k=1}^{n} x_{k}^{2}\right) a+\left(\sum_{k=1}^{n} x_{k}\right) b=\sum_{k=1}^{n} x_{k} y_{k}$$
Use these equations to find the least squares line for the following data points.
$$(1,3), \quad(2,5), \quad(3,6), \quad(5,6), \quad(7,9)$$
Sketch the points and your line to confirm that the line fits these points well. If your calculator computes regression lines, see whether it gives you the same line as the formulas.

AG
Ankit G.
Numerade Educator