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Algebra and Trigonometry

Michael Sullivan

Chapter 12

Systems of Equations and Inequalities - all with Video Answers

Educators

Section 1

Systems of Linear Equations: Substitution and Elimination

00:59

Problem 1

Solve the equation: $3 x+4=8-x$

Brandon Fox
Brandon Fox
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02:05

Problem 2

(a) Graph the line: $3 x+4 y=12$
(b) What is the slope of a line parallel to this line?

Brandon Fox
Brandon Fox
Numerade Educator
00:17

Problem 3

If a system of equations has no solution, it is said to be ________.

Brandon Fox
Brandon Fox
Numerade Educator
00:32

Problem 4

If a system of equations has one solution, the system is _________ and the equations are __________.

Brandon Fox
Brandon Fox
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00:10

Problem 5

If the only solution to a system of two linear equations containing two unknowns is $x=3, y=-2,$ then the lines intersect at the point __________.

Brandon Fox
Brandon Fox
Numerade Educator
00:32

Problem 6

If the lines that make up a system of two linear equations are coincident, then the system is ________ and the equations are __________.

Brandon Fox
Brandon Fox
Numerade Educator
00:25

Problem 7

If a system of two linear equations in two variables is inconsistent, which of the following best describes the graphs of the lines in the system?
(b) parallel
(a) intersecting
(c) coincident
(d) perpendicular

Brandon Fox
Brandon Fox
Numerade Educator
00:28

Problem 8

If a system of dependent equations containing three variables has the general solution $\{(x, y, z) | x=-z+4, y=-2 z+5, z \text { is any real number }\}$ which of the following is one of the infinite number of solutions of the system?
(a) $(1,-1,3)$
(b) $(0,4,5) \quad$ (c) $(4,-3,0)$
(d) $(-1,5,7)$

Brandon Fox
Brandon Fox
Numerade Educator
00:53

Problem 9

Verify that the values of the variables listed are solutions of the system of equations.
$$
\begin{aligned}
&\left\{\begin{array}{l}
{2 x-y=5} \\
{5 x+2 y=8}
\end{array}\right.\\
&x=2, y=-1 ;(2,-1)
\end{aligned}
$$

Brandon Fox
Brandon Fox
Numerade Educator
00:49

Problem 10

Verify that the values of the variables listed are solutions of the system of equations.
$$
\begin{aligned}
&\left\{\begin{aligned}
3 x+2 y &=2 \\
x-7 y &=-30
\end{aligned}\right.\\
&x=-2, y=4 ;(-2,4)
\end{aligned}
$$

Brandon Fox
Brandon Fox
Numerade Educator
00:57

Problem 11

Verify that the values of the variables listed are solutions of the system of equations.
$$
\begin{aligned}
&\left\{\begin{array}{l}
{3 x-4 y=4} \\
{\frac{1}{2} x-3 y=-\frac{1}{2}}
\end{array}\right.\\
&x=2, y=\frac{1}{2} ;\left(2, \frac{1}{2}\right)
\end{aligned}
$$

Brandon Fox
Brandon Fox
Numerade Educator
01:17

Problem 12

Verify that the values of the variables listed are solutions of the system of equations.
$$
\begin{aligned}
&\left\{\begin{array}{l}
{2 x+\frac{1}{2} y=0} \\
{3 x-4 y=-\frac{19}{2}}
\end{array}\right.\\
&x=-\frac{1}{2}, y=2 ;\left(-\frac{1}{2}, 2\right)
\end{aligned}
$$

Brandon Fox
Brandon Fox
Numerade Educator
00:30

Problem 13

Verify that the values of the variables listed are solutions of the system of equations.$$
\begin{aligned}
&\left\{\begin{array}{l}
{x-y=3} \\
{\frac{1}{2} x+y=3}
\end{array}\right.\\
&x=4, y=1 ;(4,1)
\end{aligned}
$$

Brandon Fox
Brandon Fox
Numerade Educator
00:41

Problem 14

Verify that the values of the variables listed are solutions of the system of equations.
$$
\begin{aligned}
&\left\{\begin{array}{r}
{x-y=3} \\
{-3 x+y=1}
\end{array}\right.\\
&x=-2, y=-5 ;(-2,-5)
\end{aligned}
$$

Brandon Fox
Brandon Fox
Numerade Educator
01:19

Problem 15

Verify that the values of the variables listed are solutions of the system of equations.
$$
\begin{aligned}
&\left\{\begin{array}{r}
{3 x+3 y+2 z=4} \\
{x-y-z=0} \\
{2 y-3 z=-8}
\end{array}\right.\\
&x=1, y=-1, z=2\\
&(1,-1,2)
\end{aligned}
$$

Brandon Fox
Brandon Fox
Numerade Educator
01:10

Problem 16

Verify that the values of the variables listed are solutions of the system of equations.
$$
\begin{aligned}
&\left\{\begin{aligned}
4 x &-z=7 \\
8 x+5 y-z &=0 \\
-x-y+5 z &=6
\end{aligned}\right.\\
&x=2, y=-3, z=1\\
&(2,-3,1)
\end{aligned}
$$

Brandon Fox
Brandon Fox
Numerade Educator
01:32

Problem 17

Verify that the values of the variables listed are solutions of the system of equations.
$$
\begin{aligned}
&\left\{\begin{aligned}
3 x+3 y+2 z &=4 \\
x-3 y+z &=10 \\
5 x-2 y-3 z &=8
\end{aligned}\right.\\
&x=2, y=-2, z=2 ;(2,-2,2)
\end{aligned}
$$

Brandon Fox
Brandon Fox
Numerade Educator
01:15

Problem 18

Verify that the values of the variables listed are solutions of the system of equations.
$$
\begin{aligned}
&\left\{\begin{aligned}
4 x-5 z &=6 \\
5 y-z &=-17 \\
-x-6 y+5 z &=24
\end{aligned}\right.\\
&x=4, y=-3, z=2 ;(4,-3,2)
\end{aligned}
$$

Brandon Fox
Brandon Fox
Numerade Educator
01:59

Problem 19

In Problems $19-56,$ solve each system of equations. If the system has no solution, say that it is inconsistent. For Problems $19-30,$ graph the lines of the system.
$$
\left\{\begin{array}{l}
{x+y=8} \\
{x-y=4}
\end{array}\right.
$$

Brandon Fox
Brandon Fox
Numerade Educator
01:57

Problem 20

In Problems $19-56,$ solve each system of equations. If the system has no solution, say that it is inconsistent. For Problems $19-30,$ graph the lines of the system.
$$
\left\{\begin{array}{l}
{x+2 y=-7} \\
{x+y=-3}
\end{array}\right.
$$

Brandon Fox
Brandon Fox
Numerade Educator
02:30

Problem 21

In Problems $19-56,$ solve each system of equations. If the system has no solution, say that it is inconsistent. For Problems $19-30,$ graph the lines of the system.
$$
\left\{\begin{array}{l}
{5 x-y=21} \\
{2 x+3 y=-12}
\end{array}\right.
$$

Brandon Fox
Brandon Fox
Numerade Educator
01:57

Problem 22

In Problems $19-56,$ solve each system of equations. If the system has no solution, say that it is inconsistent. For Problems $19-30,$ graph the lines of the system.
$$
\left\{\begin{aligned}
x+3 y &=5 \\
2 x-3 y &=-8
\end{aligned}\right.
$$

Brandon Fox
Brandon Fox
Numerade Educator
01:31

Problem 23

In Problems $19-56,$ solve each system of equations. If the system has no solution, say that it is inconsistent. For Problems $19-30,$ graph the lines of the system.
$$
\left\{\begin{aligned}
3 x &=24 \\
x+2 y &=0
\end{aligned}\right.
$$

Brandon Fox
Brandon Fox
Numerade Educator
02:07

Problem 24

In Problems $19-56,$ solve each system of equations. If the system has no solution, say that it is inconsistent. For Problems $19-30,$ graph the lines of the system.
$$
\left\{\begin{aligned}
4 x+5 y &=-3 \\
-2 y &=-8
\end{aligned}\right.
$$

Brandon Fox
Brandon Fox
Numerade Educator
02:59

Problem 25

In Problems $19-56,$ solve each system of equations. If the system has no solution, say that it is inconsistent. For Problems $19-30,$ graph the lines of the system.
$$
\left\{\begin{array}{l}
{3 x-6 y=2} \\
{5 x+4 y=1}
\end{array}\right.
$$

Dharmendra Jain
Dharmendra Jain
Numerade Educator
02:31

Problem 26

In Problems $19-56,$ solve each system of equations. If the system has no solution, say that it is inconsistent. For Problems $19-30,$ graph the lines of the system.
$$
\left\{\begin{array}{l}
{2 x+4 y=\frac{2}{3}} \\
{3 x-5 y=-10}
\end{array}\right.
$$

Dharmendra Jain
Dharmendra Jain
Numerade Educator
01:04

Problem 27

In Problems $19-56,$ solve each system of equations. If the system has no solution, say that it is inconsistent. For Problems $19-30,$ graph the lines of the system.
$$
\left\{\begin{array}{l}
{2 x+y=1} \\
{4 x+2 y=3}
\end{array}\right.
$$

Dharmendra Jain
Dharmendra Jain
Numerade Educator
01:02

Problem 28

In Problems $19-56,$ solve each system of equations. If the system has no solution, say that it is inconsistent. For Problems $19-30,$ graph the lines of the system.
$$
\left\{\begin{array}{r}
{x-y=5} \\
{-3 x+3 y=2}
\end{array}\right.
$$

Dharmendra Jain
Dharmendra Jain
Numerade Educator
01:24

Problem 29

In Problems $19-56,$ solve each system of equations. If the system has no solution, say that it is inconsistent. For Problems $19-30,$ graph the lines of the system.
$$
\left\{\begin{array}{l}
{2 x-y=0} \\
{4 x+2 y=12}
\end{array}\right.
$$

Dharmendra Jain
Dharmendra Jain
Numerade Educator
02:09

Problem 30

In Problems $19-56,$ solve each system of equations. If the system has no solution, say that it is inconsistent. For Problems $19-30,$ graph the lines of the system.
$$
\left\{\begin{array}{l}
{3 x+3 y=-1} \\
{4 x+y=\frac{8}{3}}
\end{array}\right.
$$

Dharmendra Jain
Dharmendra Jain
Numerade Educator
00:44

Problem 31

In Problems $19-56,$ solve each system of equations. If the system has no solution, say that it is inconsistent. For Problems $19-30,$ graph the lines of the system.
$$
\left\{\begin{array}{r}
{x+2 y=4} \\
{2 x+4 y=8}
\end{array}\right.
$$

Brandon Fox
Brandon Fox
Numerade Educator
00:42

Problem 32

In Problems $19-56,$ solve each system of equations. If the system has no solution, say that it is inconsistent. For Problems $19-30,$ graph the lines of the system.
$$
\left\{\begin{array}{l}
{3 x-y=7} \\
{9 x-3 y=21}
\end{array}\right.
$$

Brandon Fox
Brandon Fox
Numerade Educator
00:48

Problem 33

In Problems $19-56,$ solve each system of equations. If the system has no solution, say that it is inconsistent. For Problems $19-30,$ graph the lines of the system.
$$
\left\{\begin{aligned}
2 x-3 y &=-1 \\
10 x+y &=11
\end{aligned}\right.
$$

Brandon Fox
Brandon Fox
Numerade Educator
01:01

Problem 34

In Problems $19-56,$ solve each system of equations. If the system has no solution, say that it is inconsistent. For Problems $19-30,$ graph the lines of the system.
$$
\left\{\begin{array}{l}
{3 x-2 y=0} \\
{5 x+10 y=4}
\end{array}\right.
$$

Brandon Fox
Brandon Fox
Numerade Educator
00:51

Problem 35

In Problems $19-56,$ solve each system of equations. If the system has no solution, say that it is inconsistent. For Problems $19-30,$ graph the lines of the system.
$$
\left\{\begin{array}{r}
{2 x+3 y=6} \\
{x-y=\frac{1}{2}}
\end{array}\right.
$$

Brandon Fox
Brandon Fox
Numerade Educator
00:44

Problem 36

In Problems $19-56,$ solve each system of equations. If the system has no solution, say that it is inconsistent. For Problems $19-30,$ graph the lines of the system.
$$
\left\{\begin{array}{r}
{\frac{1}{2} x+y=-2} \\
{x-2 y=8}
\end{array}\right.
$$

Brandon Fox
Brandon Fox
Numerade Educator
01:08

Problem 37

In Problems $19-56,$ solve each system of equations. If the system has no solution, say that it is inconsistent. For Problems $19-30,$ graph the lines of the system.
$$
\left\{\begin{array}{l}
{\frac{1}{2} x+\frac{1}{3} y=3} \\
{\frac{1}{4} x-\frac{2}{3} y=-1}
\end{array}\right.
$$

Brandon Fox
Brandon Fox
Numerade Educator
01:52

Problem 38

In Problems $19-56,$ solve each system of equations. If the system has no solution, say that it is inconsistent. For Problems $19-30,$ graph the lines of the system.
$$
\left\{\begin{array}{l}
{\frac{1}{3} x-\frac{3}{2} y=-5} \\
{\frac{3}{4} x+\frac{1}{3} y=11}
\end{array}\right.
$$

Brandon Fox
Brandon Fox
Numerade Educator
01:12

Problem 39

In Problems $19-56,$ solve each system of equations. If the system has no solution, say that it is inconsistent. For Problems $19-30,$ graph the lines of the system.
$$
\left\{\begin{array}{c}
{3 x-5 y=3} \\
{15 x+5 y=21}
\end{array}\right.
$$

Brandon Fox
Brandon Fox
Numerade Educator
00:46

Problem 40

In Problems $19-56,$ solve each system of equations. If the system has no solution, say that it is inconsistent. For Problems $19-30,$ graph the lines of the system.
$$
\left\{\begin{array}{r}
{2 x-y=-1} \\
{x+\frac{1}{2} y=\frac{3}{2}}
\end{array}\right.
$$

Brandon Fox
Brandon Fox
Numerade Educator
01:19

Problem 41

In Problems $19-56,$ solve each system of equations. If the system has no solution, say that it is inconsistent. For Problems $19-30,$ graph the lines of the system.
$$
\left\{\begin{array}{l}
{\frac{1}{x}+\frac{1}{y}=8} \\
{\frac{3}{x}-\frac{5}{y}=0}
\end{array}\right.
$$

Dharmendra Jain
Dharmendra Jain
Numerade Educator
01:14

Problem 42

In Problems $19-56,$ solve each system of equations. If the system has no solution, say that it is inconsistent. For Problems $19-30,$ graph the lines of the system.
$$
\left\{\begin{array}{l}
{\frac{4}{x}-\frac{3}{y}=0} \\
{\frac{6}{x}+\frac{3}{2 y}=2}
\end{array}\right.
$$

Dharmendra Jain
Dharmendra Jain
Numerade Educator
02:27

Problem 43

In Problems $19-56,$ solve each system of equations. If the system has no solution, say that it is inconsistent. For Problems $19-30,$ graph the lines of the system.
$$
\left\{\begin{aligned}
x-y &=6 \\
2 x &-3 z=16 \\
2 y+z &=4
\end{aligned}\right.
$$

Dharmendra Jain
Dharmendra Jain
Numerade Educator
02:21

Problem 44

In Problems $19-56,$ solve each system of equations. If the system has no solution, say that it is inconsistent. For Problems $19-30,$ graph the lines of the system.
$$
\left\{\begin{aligned}
2 x+y &=-4 \\
-2 y+4 z &=0 \\
3 x &-2 z=-11
\end{aligned}\right.
$$

Dharmendra Jain
Dharmendra Jain
Numerade Educator
02:49

Problem 45

In Problems $19-56,$ solve each system of equations. If the system has no solution, say that it is inconsistent. For Problems $19-30,$ graph the lines of the system.
$$
\left\{\begin{aligned}
x-2 y+3 z &=7 \\
2 x+y+z &=4 \\
-3 x+2 y-2 z &=-10
\end{aligned}\right.
$$

Dharmendra Jain
Dharmendra Jain
Numerade Educator
02:52

Problem 46

In Problems $19-56,$ solve each system of equations. If the system has no solution, say that it is inconsistent. For Problems $19-30,$ graph the lines of the system.
$$
\left\{\begin{array}{r}
{2 x+y-3 z=0} \\
{-2 x+2 y+z=-7} \\
{3 x-4 y-3 z=7}
\end{array}\right.
$$

Dharmendra Jain
Dharmendra Jain
Numerade Educator
01:08

Problem 47

In Problems $19-56,$ solve each system of equations. If the system has no solution, say that it is inconsistent. For Problems $19-30,$ graph the lines of the system.
$$
\left\{\begin{array}{r}
{x-y-z=1} \\
{2 x+3 y+z=2} \\
{3 x+2 y=0}
\end{array}\right.
$$

Dharmendra Jain
Dharmendra Jain
Numerade Educator
01:27

Problem 48

In Problems $19-56,$ solve each system of equations. If the system has no solution, say that it is inconsistent. For Problems $19-30,$ graph the lines of the system.
$$
\left\{\begin{aligned}
2 x-3 y-z &=0 \\
-x+2 y+z &=5 \\
3 x-4 y-z &=1
\end{aligned}\right.
$$

Dharmendra Jain
Dharmendra Jain
Numerade Educator
02:10

Problem 49

In Problems $19-56,$ solve each system of equations. If the system has no solution, say that it is inconsistent. For Problems $19-30,$ graph the lines of the system.
$$
\left\{\begin{array}{r}
{x-y-z=1} \\
{-x+2 y-3 z=-4} \\
{3 x-2 y-7 z=0}
\end{array}\right.
$$

Dharmendra Jain
Dharmendra Jain
Numerade Educator
01:52

Problem 50

In Problems $19-56,$ solve each system of equations. If the system has no solution, say that it is inconsistent. For Problems $19-30,$ graph the lines of the system.
$$
\left\{\begin{aligned}
2 x-3 y-z &=0 \\
3 x+2 y+2 z &=2 \\
x+5 y+3 z &=2
\end{aligned}\right.
$$

Dharmendra Jain
Dharmendra Jain
Numerade Educator
01:58

Problem 51

In Problems $19-56,$ solve each system of equations. If the system has no solution, say that it is inconsistent. For Problems $19-30,$ graph the lines of the system.
$$
\left\{\begin{array}{r}
{2 x-2 y+3 z=6} \\
{4 x-3 y+2 z=0} \\
{-2 x+3 y-7 z=1}
\end{array}\right.
$$

Dharmendra Jain
Dharmendra Jain
Numerade Educator
01:36

Problem 52

In Problems $19-56,$ solve each system of equations. If the system has no solution, say that it is inconsistent. For Problems $19-30,$ graph the lines of the system.
$$
\left\{\begin{array}{l}
{3 x-2 y+2 z=6} \\
{7 x-3 y+2 z=-1} \\
{2 x-3 y+4 z=0}
\end{array}\right.
$$

Dharmendra Jain
Dharmendra Jain
Numerade Educator
01:45

Problem 53

In Problems $19-56,$ solve each system of equations. If the system has no solution, say that it is inconsistent. For Problems $19-30,$ graph the lines of the system.
$$
\left\{\begin{aligned}
x+y-z &=6 \\
3 x-2 y+z &=-5 \\
x+3 y-2 z &=14
\end{aligned}\right.
$$

Dharmendra Jain
Dharmendra Jain
Numerade Educator
02:26

Problem 54

In Problems $19-56,$ solve each system of equations. If the system has no solution, say that it is inconsistent. For Problems $19-30,$ graph the lines of the system.
$$
\left\{\begin{aligned}
x-y+z=&-4 \\
2 x-3 y+4 z=&-15 \\
5 x+y-2 z=& 12
\end{aligned}\right.
$$

Dharmendra Jain
Dharmendra Jain
Numerade Educator
02:03

Problem 55

In Problems $19-56,$ solve each system of equations. If the system has no solution, say that it is inconsistent. For Problems $19-30,$ graph the lines of the system.
$$
\left\{\begin{array}{r}
{x+2 y-z=-3} \\
{2 x-4 y+z=-7} \\
{-2 x+2 y-3 z=4}
\end{array}\right.
$$

Dharmendra Jain
Dharmendra Jain
Numerade Educator
01:57

Problem 56

In Problems $19-56,$ solve each system of equations. If the system has no solution, say that it is inconsistent. For Problems $19-30,$ graph the lines of the system.
$$
\left\{\begin{aligned}
x+4 y-3 z &=-8 \\
3 x-y+3 z &=12 \\
x+y+6 z &=1
\end{aligned}\right.
$$

Dharmendra Jain
Dharmendra Jain
Numerade Educator
01:13

Problem 57

The perimeter of a rectangular floor is 90 feet. Find the dimensions of the floor if the length is twice the width.

Brandon Fox
Brandon Fox
Numerade Educator
02:14

Problem 58

The length of fence required to enclose a rectangular field is 3000 meters. What are the dimensions of the field if it is known that the difference between its length and width is 50 meters?

Brandon Fox
Brandon Fox
Numerade Educator
01:26

Problem 59

In 2013 there was a total of 81 commercial and noncommercial orbital launches worldwide. In addition, the number of noncommercial orbital launches was twelve more than twice the number of commercial orbital launches. Determine the number of commercial and noncommercial orbital launches in 2013 .

Brandon Fox
Brandon Fox
Numerade Educator
01:40

Problem 60

Tickets A movie theater charges $\$ 9.00$ for adults and $\$ 700$ for senior citizens. On a day when 325 people paid for admission, the total receipts were $\$ 2495 .$ How many who paid were adults? How many were seniors?

Brandon Fox
Brandon Fox
Numerade Educator
01:54

Problem 61

A store sells cashews for $\$ 5.00$ per pound and peanuts for $\$ 1.50$ per pound. The manager decides to mix 30 pounds of peanuts with some cashews and sell the mixture for $\$ 3.00$ per pound. How many pounds of cashews should be mixed with the peanuts so that the mixture will produce the same revenue as selling the nuts separately?

Brandon Fox
Brandon Fox
Numerade Educator
02:29

Problem 62

A chemist wants to make 14 liters of a $40 \%$ acid solution. She has solutions that are $30 \%$ acid and $65 \%$ acid. How much of each must she mix?

Dharmendra Jain
Dharmendra Jain
Numerade Educator
02:41

Problem 63

A wireless store owner takes presale orders for a new smartphone and tablet. He gets 340 preorders for the smartphone and 250 preorders for the tablet. The combined value of the preorders is $\$ 270,500 .$ If the price of a smartphone and tablet together is $\$ 965,$ how much does each device cost?

Brandon Fox
Brandon Fox
Numerade Educator
02:15

Problem 64

Financial Planning A recently retired couple needs $\$ 12,000$ per year to supplement their Social Security. They have $\$ 150,000$ to invest to obtain this income. They have decided on two investment options: AA bonds yielding $10 \%$ per annum and a Bank Certificate yielding $5 \% .$
(a) How much should be invested in each to realize exactly $\$ 12,000 ?$
(b) If, after 2 years, the couple requires $\$ 14,000$ per year in income, how should they reallocate their investment to achieve the new amount?

Brandon Fox
Brandon Fox
Numerade Educator
02:14

Problem 65

With a tail wind, a small Piper aircraft can fly 600 miles in 3 hours. Against this same wind, the Piper can fly the same distance in 4 hours. Find the average wind speed and the average airspeed of the Piper.

Dharmendra Jain
Dharmendra Jain
Numerade Educator
01:39

Problem 66

The average airspeed of a single engine aircraft is 150 miles per hour. If the aircraft flew the same distance in 2 hours with the wind as it flew in 3 hours against the wind, what was the wind speed?

Dharmendra Jain
Dharmendra Jain
Numerade Educator
02:15

Problem 67

A restaurant manager wants to purchase 200 sets of dishes. One design costs $\$ 25$ per set, and another costs $\$ 45$ per set. If she has only $\$ 7400$ to spend, how many sets of each design should she order?

Brandon Fox
Brandon Fox
Numerade Educator
02:57

Problem 68

One group of people purchased 10 hot dogs and 5 soft drinks at a cost of $\$ 35.00 .$ A second bought 7 hot dogs and 4 soft drinks at a cost of $\$ 25.25 .$ What is the cost of a single hot dog? A single soft drink?

Brandon Fox
Brandon Fox
Numerade Educator
03:35

Problem 69

The grocery store we use does not mark prices on its goods. My wife went to this store, bought three 1 -pound packages of bacon and two cartons of eggs, and paid a total of $\$ 13.45 .$ Not knowing that she went to the store, I also went to the same store, purchased two 1 -pound packages of bacon and three cartons of eggs, and paid a total of $\$ 11.45 .$ Now we want to return two 1-pound packages of bacon and two cartons of eggs. How much will be refunded?

Brandon Fox
Brandon Fox
Numerade Educator
02:01

Problem 70

Pamela requires 3 hours to swim 15 miles downstream on the Illinois River. The return trip upstream takes 5 hours. Find Pamela's average speed in still water. How fast is the current? (Assume that Pamela's speed is the same in each direction.)

Dharmendra Jain
Dharmendra Jain
Numerade Educator
02:32

Problem 71

A doctor's prescription calls for a daily intake containing 40 milligrams (mg) of vitamin $\mathrm{C}$ and $30 \mathrm{mg}$ of vitamin D. Your pharmacy stocks two liquids that can be used: One contains $20 \%$ vitamin $\mathrm{C}$ and $30 \%$ vitamin $\mathrm{D}$ the other $40 \%$ vitamin $\mathrm{C}$ and $20 \%$ vitamin D. How many milligrams of each compound should be mixed to fill the prescription?

Dharmendra Jain
Dharmendra Jain
Numerade Educator
01:49

Problem 72

A doctor's prescription calls for the creation of pills that contain 12 units of vitamin $\mathrm{B}_{12}$ and 12 units of vitamin E. Your pharmacy stocks two powders that can be used to make these pills: One contains $20 \%$ vitamin $\mathrm{B}_{12}$ and $30 \%$ vitamin $\mathrm{E},$ the other $40 \%$ vitamin $\mathrm{B}_{12}$ and $20 \%$ vitamin E. How many units of each powder should be mixed in each pill?

Dharmendra Jain
Dharmendra Jain
Numerade Educator
06:10

Problem 73

Find real numbers $a, b,$ and $c$ so that the graph of the function $y=a x^{2}+b x+c$ contains the points $(-1,4),(2,3),$ and $(0,1)$

Kelly Hughes
Kelly Hughes
Numerade Educator
07:19

Problem 74

Find real numbers $a, b,$ and $c$ so that the graph of the function $y=a x^{2}+b x+c$ contains the points $(-1,-2),(1,-4),$ and $(2,4)$

Kelly Hughes
Kelly Hughes
Numerade Educator
01:36

Problem 75

In economics, the IS curve is a linear equation that represents all combinations of income $Y$ and interest rates $r$ that maintain an equilibrium in the market for goods in the economy. The LM curve is a linear equation that represents all combinations of income $Y$ and interest rates $r$ that maintain an equilibrium in the market for money in the economy. In an economy, suppose that the equilibrium level of income (in millions of dollars) and interest rates satisfy the system of equations
$$
\left\{\begin{array}{l}
{0.06 Y-5000 r=240} \\
{0.06 Y+6000 r=900}
\end{array}\right.
$$
Find the equilibrium level of income and interest rates.

Brandon Fox
Brandon Fox
Numerade Educator
01:09

Problem 76

In economics, the IS curve is a linear equation that represents all combinations of income $Y$ and interest rates $r$ that maintain an equilibrium in the market for goods in the economy. The LM curve is a linear equation that represents all combinations of income $Y$ and interest rates $r$ that maintain an equilibrium in the market for money in the economy. In an economy, suppose that the equilibrium level of income (in millions of dollars) and interest rates satisfy the system of equations
$$
\left\{\begin{array}{l}
{0.05 Y-1000 r=10} \\
{0.05 Y+800 r=100}
\end{array}\right.
$$
Find the equilibrium level of income and interest rates.

Brandon Fox
Brandon Fox
Numerade Educator
02:20

Problem 77

Electricity: Kirchhoff's Rules An application of Kirchhoff's Rules to the circuit shown results in the following system of equations:
$$
\left\{\begin{aligned}
I_{2} &=I_{1}+I_{3} \\
5-3 I_{1}-5 I_{2} &=0 \\
10-5 I_{2}-7 I_{3} &=0
\end{aligned}\right.
$$
Find the currents $I_{1}, I_{2},$ and $I_{3}$.
(IMAGE CAN'T COPY)

Dharmendra Jain
Dharmendra Jain
Numerade Educator
02:18

Problem 78

Electricity: Kirchhoffs Rules An application of Kirchhoff's Rules to the circuit shown below results in the following system of equations:
$$
\left\{\begin{aligned}
I_{3} &=I_{1}+I_{2} \\
8 &=4 I_{3}+6 I_{2} \\
8 I_{1} &=4+6 I_{2}
\end{aligned}\right.
$$
Find the currents $I_{1}, I_{2},$ and $I_{3}$
(IMAGE CAN'T COPY)

Dharmendra Jain
Dharmendra Jain
Numerade Educator
02:27

Problem 79

A Broadway theater has 500 seats, divided into orchestra, main, and balcony seating. Orchestra seats sell for $\$ 150,$ main seats for $\$ 135,$ and balcony seats for $\$ 110 .$ If all the seats are sold, the gross revenue to the theater is $\$ 64,250 .$ If all the main and balcony seats are sold, but only half the orchestra seats are sold, the gross revenue is $\$ 56,750$. How many of each kind of seat are there?

Dharmendra Jain
Dharmendra Jain
Numerade Educator
02:14

Problem 80

A movie theater charges $\$ 11.00$ for adults, $\$ 6.50$ for children, and $\$ 9.00$ for senior citizens. One day the theater sold 405 tickets and collected $\$ 3315$ in receipts. Twice as many children's tickets were sold as adult tickets. How many adults, children, and senior citizens went to the theater that day?

Dharmendra Jain
Dharmendra Jain
Numerade Educator
03:32

Problem 81

A dietitian wishes a patient to have a meal that has 66 grams (g) of protein, 94.5 g of carbohydrates, and 910 milligrams (mg) of calcium. The hospital food service tells the dietitian that the dinner for today is chicken, corn, and $2 \%$ milk. Each serving of chicken has $30 \mathrm{g}$ of protein,
$35 \mathrm{g}$ of carbohydrates, and $200 \mathrm{mg}$ of calcium. Each serving of corn has 3 g of protein, $16 \mathrm{g}$ of carbohydrates, and $10 \mathrm{mg}$ of calcium. Each glass of $2 \%$ milk has 9 g of protein, 13 g of carbohydrates, and $300 \mathrm{mg}$ of calcium. How many servings of each food should the dietitian provide for the patient?

Dharmendra Jain
Dharmendra Jain
Numerade Educator
03:02

Problem 82

Kelly has $\$ 20,000$ to invest. As her financial planner, you recommend that she diversify into three investments: Treasury bills that yield $5 \%$ simple interest, Treasury bonds that yield $7 \%$ simple interest, and corporate bonds that yield $10 \%$ simple interest. Kelly wishes to earn $\$ 1390$ per year in income. Also, Kelly wants her investment in Treasury bills to be $\$ 3000$ more than her investment in corporate bonds. How much money should Kelly place in each investment?

Dharmendra Jain
Dharmendra Jain
Numerade Educator
04:59

Problem 83

One group of customers bought 8 deluxe hamburgers, 6 orders of large fries, and 6 large colas for $\$ 26.10 .$ A second group ordered 10 deluxe hamburgers, 6 large fries, and 8 large colas and paid $\$ 31.60 .$ Is there sufficient information to determine the price of each food item? If not, construct a table showing the various possibilities. Assume that the hamburgers cost between $\$ 1.75$ and $\$ 2.25,$ the fries between $\$ 0.75$ and $\$ 1.00,$ and the colas between $\$ 0.60$ and $\$ 0.90$

Darssan Eswaramoorthi
Darssan Eswaramoorthi
River Bluff High School
02:30

Problem 84

Use the information given in Problem 83 . Suppose that a third group purchased 3 deluxe hamburgers, 2 large fries, and 4 large colas for $\$ 10.95 .$ Now is there sufficient information to determine the price of each food item? If so, determine each price.

Dharmendra Jain
Dharmendra Jain
Numerade Educator
03:10

Problem 85

Three painters (Beth, Bill, and Edie), working together, can paint the exterior of a home in 10 hours (h). Bill and Edie together have painted a similar house in 15 h. One day, all three worked on this same kind of house for 4 h, after which Edie left. Beth and Bill required 8 more hours to finish. Assuming no gain or loss in efficiency, how long should it take each person to complete such a job alone?

Dharmendra Jain
Dharmendra Jain
Numerade Educator
02:27

Problem 86

Make up a system of three linear equations containing three variables that has:
(a) No solution.
(b) Exactly one solution.
(c) Infinitely many solutions.
Give the three systems to a friend to solve and critique.

Dharmendra Jain
Dharmendra Jain
Numerade Educator
02:15

Problem 87

Write a brief paragraph outlining your strategy for solving a system of two linear equations containing two variables.

Dharmendra Jain
Dharmendra Jain
Numerade Educator
02:09

Problem 88

Do you prefer the method of substitution or the method of elimination for solving a system of two linear equations containing two variables? Give your reasons.

Brandon Fox
Brandon Fox
Numerade Educator
View

Problem 89

$$
\text { Graph } f(x)=-3^{1-x}+2
$$

Kathleen Carty
Kathleen Carty
Numerade Educator
02:36

Problem 90

Factor each of the following:
(a) $4(2 x-3)^{3} \cdot 2 \cdot\left(x^{3}+5\right)^{2}+2\left(x^{3}+5\right) \cdot 3 x^{2} \cdot(2 x-3)^{4}$
(b) $\frac{1}{2}(3 x-5)^{-\frac{1}{2}} \cdot 3 \cdot(x+3)^{-\frac{1}{2}}-\frac{1}{2}(x+3)^{-\frac{3}{2}}(3 x-5)^{\frac{1}{2}}$

Jennifer Stoner
Jennifer Stoner
Numerade Educator
01:27

Problem 91

Find the exact value of $\sin ^{-1}\left[\sin \left(-\frac{10 \pi}{9}\right)\right]$

Subhadeepta Sahoo
Subhadeepta Sahoo
Numerade Educator
01:14

Problem 92

Write $-\sqrt{3}+i$ in polar form. Express each argument in degrees.

Dharmendra Jain
Dharmendra Jain
Numerade Educator