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Algebra and Trigonometry

James Stewart, Lothar Redlin, Saleem Watson

Chapter 10

Systems of Equations and Inequalities - all with Video Answers

Educators

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Section 1

Systems of Linear Equations in Two Variables

02:06

Problem 1

The system of equations

$$\left\{\begin{aligned} 2 x+3 y &=7 \\ 5 x-y &=9 \end{aligned}\right.$$

is a system of two equations in the two variables_____and_____ $\longrightarrow$ To determine whether $(5,-1)$ is a solution of this system, we check whether $x=5$ and $y=-1$ satisfy each $$\frac{\text { in the system. Which of the following are }}{\text { solutions of this system? }}$$

$(5,-1), \quad(-1,3), \quad(2,1)$

Lynn Larson
Lynn Larson
Numerade Educator
00:28

Problem 2

A system of equations in two variables can be solved by the_____method, the_____method, or the_____method.

Amy Jiang
Amy Jiang
Numerade Educator
00:31

Problem 3

A system of two linear equations in two variables can have one solution,_____solution, or_____solutions.

James Macpherson
James Macpherson
Numerade Educator
02:30

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 $$(1, \longrightarrow),(-3, \longrightarrow),$$and $(5, \longrightarrow)$$

Amy Jiang
Amy Jiang
Numerade Educator
01:43

Problem 5

$5-8=$ Substitution Method Use the substitution method to find
all solutions of the system of equations.

$\left\{\begin{aligned} x-y &=1 \\ 4 x+3 y &=18 \end{aligned}\right.$

James Macpherson
James Macpherson
Numerade Educator
01:39

Problem 6

$5-8=$ Substitution Method Use the substitution method to find
all solutions of the system of equations.

$3 x+y=1$
$5 x+2 y=1$

Amy Jiang
Amy Jiang
Numerade Educator
01:15

Problem 7

$5-8=$ Substitution Method Use the substitution method to find
all solutions of the system of equations.

$x-y=2$
$2 x+3 y=9$

James Macpherson
James Macpherson
Numerade Educator
01:57

Problem 8

$5-8=$ Substitution Method Use the substitution method to find
all solutions of the system of equations.

$2 x+y=7$
$x+2 y=2$

Amy Jiang
Amy Jiang
Numerade Educator
01:18

Problem 9

$9-12=$ Elimination Method Use the elimination method to find all solutions of the system of equations.

$\left\{\begin{aligned} 3 x+4 y &=10 \\ x-4 y &=-2 \end{aligned}\right.$

James Macpherson
James Macpherson
Numerade Educator
01:44

Problem 10

$9-12=$ Elimination Method Use the elimination method to find all solutions of the system of equations.

$2 x+5 y=15$
$4 x+y=21$

Amy Jiang
Amy Jiang
Numerade Educator
02:02

Problem 11

$9-12=$ Elimination Method Use the elimination method to find all solutions of the system of equations.

$3 x-2 y=-13$
$-6 x+5 y=28$

James Macpherson
James Macpherson
Numerade Educator
01:51

Problem 12

$9-12$ . . Elimination Method Use the elimination method to find all solutions of the system of equations.

$2 x-5 y=-18$
$3 x+4 y=19$

Amy Jiang
Amy Jiang
Numerade Educator
01:58

Problem 13

$13-14=$ Graphical Method Two equations and their graphs are given. Find the intersection point(s) of the graphs by solving the system.

$\left\{\begin{aligned} 2 x+y &=-1 \\ x-2 y &=-8 \end{aligned}\right.$

James Macpherson
James Macpherson
Numerade Educator
01:27

Problem 14

$13-14=$ Graphical Method Two equations and their graphs are given. Find the intersection point(s) of the graphs by solving the system.

$x+y=2$
$2 x+y=5$

Amy Jiang
Amy Jiang
Numerade Educator
01:11

Problem 15

$15-20$ : Number of Solutions Determined Graphically 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.

$x-y=4$
$2 x+y=2$

James Macpherson
James Macpherson
Numerade Educator
00:40

Problem 16

$15-20$ : Number of Solutions Determined Graphically 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.

$2 x-y=4$
$3 x+y=6$

Amy Jiang
Amy Jiang
Numerade Educator
01:17

Problem 17

$15-20$ : Number of Solutions Determined Graphically 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.

$2 x-3 y=12$
$-x+\frac{3}{2} y=4$

James Macpherson
James Macpherson
Numerade Educator
00:44

Problem 18

$15-20$ : Number of Solutions Determined Graphically 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.

$\left\{\begin{aligned} 2 x+6 y &=0 \\-3 x-9 y &=18 \end{aligned}\right.$

Amy Jiang
Amy Jiang
Numerade Educator
01:14

Problem 19

$15-20$ : Number of Solutions Determined Graphically 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.

$\left\{\begin{aligned}-x+\frac{1}{2} y &=-5 \\ 2 x-y &=10 \end{aligned}\right.$

James Macpherson
James Macpherson
Numerade Educator
01:42

Problem 20

$15-20$ : Number of Solutions Determined Graphically 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.

$\left\{\begin{aligned} 12 x+15 y &=-18 \\ 2 x+\frac{5}{2} y &=-3 \end{aligned}\right.$

Amy Jiang
Amy Jiang
Numerade Educator
00:56

Problem 21

$21-50=$ Solving a System of Equations 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 $6 .$

$\left\{\begin{aligned} x+y &=4 \\-x+y &=0 \end{aligned}\right.$

James Macpherson
James Macpherson
Numerade Educator
01:26

Problem 22

$21-50=$ Solving a System of Equations 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 $6 .$

$2 x-3 y=9$
$4 x+3 y=9$

Amy Jiang
Amy Jiang
Numerade Educator
00:56

Problem 23

$21-50=$ Solving a System of Equations 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 $6 .$

$2 x-3 y=9$
$4 x+3 y=9$

James Macpherson
James Macpherson
Numerade Educator
02:03

Problem 24

$21-50=$ Solving a System of Equations 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 $6 .$

$3 x+2 y=0$
$-x-2 y=8$

Amy Jiang
Amy Jiang
Numerade Educator
01:40

Problem 25

$21-50=$ Solving a System of Equations 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 $6 .$

$x+3 y=5$
$2 x-y=3$

James Macpherson
James Macpherson
Numerade Educator
01:26

Problem 26

$21-50=$ Solving a System of Equations 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 $6 .$

$x+y=7$
$2 x-3 y=-1$

Amy Jiang
Amy Jiang
Numerade Educator
01:25

Problem 27

$21-50=$ Solving a System of Equations 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 $6 .$

$\left\{\begin{aligned}-x+y &=2 \\ 4 x-3 y &=-3 \end{aligned}\right.$

James Macpherson
James Macpherson
Numerade Educator
02:35

Problem 28

$21-50=$ Solving a System of Equations 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 $6 .$

$4 x-3 y=28$
$9 x-y=-6$

Amy Jiang
Amy Jiang
Numerade Educator
01:20

Problem 29

$21-50=$ Solving a System of Equations 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 $6 .$

$\left\{\begin{aligned} x+2 y &=7 \\ 5 x-y &=2 \end{aligned}\right.$

James Macpherson
James Macpherson
Numerade Educator
01:44

Problem 30

$21-50=$ Solving a System of Equations 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 $6 .$

$\left\{\begin{aligned}-4 x+12 y &=0 \\ 12 x+4 y &=160 \end{aligned}\right.$

Amy Jiang
Amy Jiang
Numerade Educator
02:09

Problem 31

$21-50=$ Solving a System of Equations 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 $6 .$

$\left\{\begin{aligned}-\frac{1}{3} x-\frac{1}{6} y &=-1 \\ \frac{2}{3} x+\frac{1}{6} y &=3 \end{aligned}\right.$

James Macpherson
James Macpherson
Numerade Educator
02:54

Problem 32

$21-50=$ Solving a System of Equations 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 $6 .$

$\left\{\begin{aligned} \frac{3}{4} x+\frac{1}{2} y &=5 \\-\frac{1}{4} x-\frac{3}{2} y &=1 \end{aligned}\right.$

Amy Jiang
Amy Jiang
Numerade Educator
02:28

Problem 33

$21-50=$ Solving a System of Equations 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 $6 .$

$\frac{1}{2} x+\frac{1}{3} y=2$
$\frac{1}{5} x-\frac{2}{3} y=8$

James Macpherson
James Macpherson
Numerade Educator
02:40

Problem 34

$21-50=$ Solving a System of Equations 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 $6 .$

$\left\{\begin{aligned} 0.2 x-0.2 y &=-1.8 \\-0.3 x+0.5 y &=3.3 \end{aligned}\right.$

Amy Jiang
Amy Jiang
Numerade Educator
01:58

Problem 35

$21-50=$ Solving a System of Equations 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 $6 .$

$\left\{\begin{aligned} 3 x+2 y &=8 \\ x-2 y &=0 \end{aligned}\right.$

James Macpherson
James Macpherson
Numerade Educator
02:11

Problem 36

$21-50=$ Solving a System of Equations 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 $6 .$

$4 x+2 y=16$
$x-5 y=70$

Amy Jiang
Amy Jiang
Numerade Educator
01:33

Problem 37

$21-50=$ Solving a System of Equations 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 $6 .$

$\left\{\begin{aligned} x+4 y &=8 \\ 3 x+12 y &=2 \end{aligned}\right.$

James Macpherson
James Macpherson
Numerade Educator
01:37

Problem 38

$21-50=$ Solving a System of Equations 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 $6 .$

$\left\{\begin{aligned}-3 x+5 y &=2 \\ 9 x-15 y &=6 \end{aligned}\right.$

Ashley Boni
Ashley Boni
Numerade Educator
01:18

Problem 39

$21-50=$ Solving a System of Equations 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 $6 .$

$\left\{\begin{aligned} 2 x-6 y &=10 \\-3 x+9 y &=-15 \end{aligned}\right.$

James Macpherson
James Macpherson
Numerade Educator
01:07

Problem 40

$21-50=$ Solving a System of Equations 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 $6 .$

$\left\{\begin{aligned} 2 x-3 y &=-8 \\ 14 x-21 y &=3 \end{aligned}\right.$

Amy Jiang
Amy Jiang
Numerade Educator
01:21

Problem 41

$21-50=$ Solving a System of Equations 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 $6 .$

$6 x+4 y=12$
$9 x+6 y=18$

James Macpherson
James Macpherson
Numerade Educator
02:38

Problem 42

$21-50=$ Solving a System of Equations 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 $6 .$

$\left\{\begin{aligned} 25 x-75 y &=100 \\-10 x+30 y &=-40 \end{aligned}\right.$

Amy Jiang
Amy Jiang
Numerade Educator
02:05

Problem 43

$21-50=$ Solving a System of Equations 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 $6 .$

$8 s-3 t=-3$
$5 s-2 t=-1$

James Macpherson
James Macpherson
Numerade Educator
01:24

Problem 44

$21-50=$ Solving a System of Equations 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 $6 .$

$u-30 v=-5$
$-3 u+80 v=5$

Amy Jiang
Amy Jiang
Numerade Educator
02:15

Problem 45

$21-50=$ Solving a System of Equations 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 $6 .$

$\left\{\begin{aligned} \frac{1}{2} x+\frac{3}{5} y &=3 \\ \frac{5}{3} x+2 y &=10 \end{aligned}\right.$

James Macpherson
James Macpherson
Numerade Educator
02:23

Problem 46

$21-50=$ Solving a System of Equations 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 $6 .$

$\left\{\begin{aligned} \frac{3}{2} x-\frac{1}{3} y &=\frac{1}{2} \\ 2 x-\frac{1}{2} y &=-\frac{1}{2} \end{aligned}\right.$

Amy Jiang
Amy Jiang
Numerade Educator
03:39

Problem 47

$21-50=$ Solving a System of Equations 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 $6 .$

$\left\{\begin{aligned} 0.4 x+1.2 y &=14 \\ 12 x-5 y &=10 \end{aligned}\right.$

James Macpherson
James Macpherson
Numerade Educator
03:00

Problem 48

$21-50=$ Solving a System of Equations 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 $6 .$

$\left\{\begin{aligned} 26 x-10 y &=-4 \\-0.6 x+1.2 y &=3 \end{aligned}\right.$

Amy Jiang
Amy Jiang
Numerade Educator
02:00

Problem 49

$21-50=$ Solving a System of Equations 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 $6 .$

$\left\{\begin{aligned} \frac{1}{3} x-\frac{1}{4} y &=2 \\-8 x+6 y &=10 \end{aligned}\right.$

James Macpherson
James Macpherson
Numerade Educator
02:05

Problem 50

$21-50=$ Solving a System of Equations 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 $6 .$

$$\left\{\begin{aligned}-\frac{1}{10} x+\frac{1}{2} y &=4 \\ 2 x-10 y &=-80 \end{aligned}\right.$$

Amy Jiang
Amy Jiang
Numerade Educator
02:00

Problem 51

$51-54=$ Solving a System of Equations Graphically 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 $[$ TRACE $]$ or by using In te $r$ se c $t$ . Round your answers to two decimals.

$0.21 x+3.17 y=9.51$
$2.35 x-1.17 y=5.89$

Amy Jiang
Amy Jiang
Numerade Educator
03:06

Problem 52

$51-54=$ Solving a System of Equations Graphically 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 $[$ TRACE $]$ or by using In te $r$ se c $t$ . Round your answers to two decimals.

$18.72 x-14.91 y=12.33$
$6.21 x-12.92 y=17.82$

Amy Jiang
Amy Jiang
Numerade Educator
03:06

Problem 53

$51-54=$ Solving a System of Equations Graphically 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 $[$ TRACE $]$ or by using In te $r$ se c $t$ . Round your answers to two decimals.

$2371 x-6552 y=13,591$
$9815 x+992 y=618,555$

Amy Jiang
Amy Jiang
Numerade Educator
02:00

Problem 54

$51-54=$ Solving a System of Equations Graphically 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 $[$ TRACE $]$ or by using In te $r$ se c $t$ . Round your answers to two decimals.

$\left\{\begin{aligned}-435 x+912 y &=0 \\ 132 x+455 y &=994 \end{aligned}\right.$

Amy Jiang
Amy Jiang
Numerade Educator
01:54

Problem 55

$55-58=$ Solving a General System of Equations Find $x$ and $y$ in terms of $a$ and $b .$

$x+y=0$ $x+a y=1$$(a \neq 1)$

James Macpherson
James Macpherson
Numerade Educator
03:06

Problem 56

$55-58=$ Solving a General System of Equations Find $x$ and $y$ in terms of $a$ and $b .$

$$\left\{\begin{aligned} a x+b y &=0 \\ x+y &=1 \end{aligned}\right.(a \neq b)$$

Amy Jiang
Amy Jiang
Numerade Educator
04:28

Problem 57

$55-58=$ Solving a General System of Equations Find $x$ and $y$ in terms of $a$ and $b .$

$a x+b y=1$ $b x+a y=1$$\left(a^{2}-b^{2} \neq 0\right)$

James Macpherson
James Macpherson
Numerade Educator
03:02

Problem 58

$55-58=$ Solving a General System of Equations Find $x$ and $y$ in terms of $a$ and $b .$

$a x+b y=0$ $a^{2} x+b^{2} y=1$$(a \neq 0, b \neq 0, a \neq b)$

Amy Jiang
Amy Jiang
Numerade Educator
01:11

Problem 59

Number Problem Find two numbers whose sum is 34 and whose difference is $10 .$

James Macpherson
James Macpherson
Numerade Educator
04:04

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.

Amy Jiang
Amy Jiang
Numerade Educator
03:19

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?

James Macpherson
James Macpherson
Numerade Educator
05:11

Problem 62

Admission Fees 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?

Amy Jiang
Amy Jiang
Numerade Educator
03:02

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?

James Macpherson
James Macpherson
Numerade Educator
04:24

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 he stand sold 135 boxes of strawberries for a total of $\$ 1110 .$ How many boxes of each type were sold?

Amy Jiang
Amy Jiang
Numerade Educator
05:15

Problem 65

Airplane Speed 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 rip takes only 1 $\mathrm{h} 12$ min. What is his speed in still air, and how fast is the wind blowing?

James Macpherson
James Macpherson
Numerade Educator
03:52

Problem 66

Boat Speed $A$ 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?

Amy Jiang
Amy Jiang
Numerade Educator
15:04

Problem 67

Nutrition 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 $\mathrm{B}$ 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?

AK
Avinash Koya
Numerade Educator
04:46

Problem 68

Coffee Blends A customer in a coffee shop purchases a blend of two coffees: Kenyan, costing $\$ 3.50$ a pound, and SriLankan, 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?

Amy Jiang
Amy Jiang
Numerade Educator
00:46

Problem 69

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

AG
Ankit Gupta
Numerade Educator
05:50

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 . of a solution that contains 14$\%$ salt?

Amy Jiang
Amy Jiang
Numerade Educator
03:41

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?

James Macpherson
James Macpherson
Numerade Educator
03:45

Problem 72

Investments $\mathrm{A}$ 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?

Amy Jiang
Amy Jiang
Numerade Educator
03:50

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 mi/h and travels 35 mi farther than Mary, who drives at 40 $\mathrm{mi} / \mathrm{h}$ . Mary's trip takes 15 min longer than John's. For what length of time does each of them drive?

James Macpherson
James Macpherson
Numerade Educator
04:22

Problem 74

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

Amy Jiang
Amy Jiang
Numerade Educator
01:42

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.

Breanna Ollech
Breanna Ollech
Numerade Educator
05:51

Problem 76

Area of a Triangle 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 $y=-4 x+20$ .

Amy Jiang
Amy Jiang
Numerade Educator
02:23

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 (see page 174). 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 $x^{\prime} s .$ See Section 13.1 for a complete description of sigma $(\Sigma)$

AG
Ankit Gupta
Numerade Educator