Algebra and Trigonometry

Michael Sullivan

Chapter 12 Systems of Equations and Inequalities - all with Video Answers

Educators

+ 4 more educators

Section 1

Systems of Linear Equations: Substitution and Elimination

00:59

Problem 1

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

Brandon Fox

Numerade Educator

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

Numerade Educator

01:41

Problem 3

True or False If a system of equations has no solution, it is said to be dependent.

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

Numerade Educator

01:16

Problem 5

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

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

Numerade Educator

01:03

Multiple Choice If a system of two linear equations in two variables is inconsistent, then the graphs of the lines in the system are ________.
(a) intersecting
(b) parallel
(c) coincident
(d) perpendicular

Brandon Fox

Numerade Educator

02:06

Problem 8

Multiple Choice If a system of dependent equations containing three variables has the general solution $\{(x, y, z) \mid x=-z+4, y=-2 z+5, z$ is any real number $\}$ then _____ is one of the infinite number of solutions of the
system.
(a) (1,-1,3)
(b) (0,4,5)
(c) (4,-3,0)
(d) (-1,5,7)

Shaikh Sadi

Numerade Educator

00:53

Problem 9

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

Brandon Fox

Numerade Educator

00:49

Problem 10

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

Brandon Fox

Numerade Educator

00:57

Problem 11

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

Brandon Fox

Numerade Educator

01:49

Problem 12

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

Brandon Fox

Numerade Educator

00:30

Problem 13

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

Brandon Fox

Numerade Educator

00:41

Problem 14

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

Brandon Fox

Numerade Educator

01:23

Problem 15

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

Kelly Hughes

Numerade Educator

01:10

Problem 16

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

Brandon Fox

Numerade Educator

01:32

Problem 17

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

Brandon Fox

Numerade Educator

01:15

Problem 18

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

Brandon Fox

Numerade Educator

01:11

Problem 19

Solve each system of equations. If the system has no solution, state that it is inconsistent. Graph the lines of the system.
$$
\left\{\begin{array}{l}
x+y=8 \\
x-y=4
\end{array}\right.
$$

Brandon Fox

Numerade Educator

01:15

Problem 20

Solve each system of equations. If the system has no solution, state that it is inconsistent. Graph the lines of the system.
$$
\left\{\begin{array}{l}
x+2 y=-7 \\
x+y=-3
\end{array}\right.
$$

Brandon Fox

Numerade Educator

03:09

Problem 21

Solve each system of equations. If the system has no solution, state that it is inconsistent. Graph the lines of the system.
$$
\left\{\begin{array}{l}
5 x-y=21 \\
2 x+3 y=-12
\end{array}\right.
$$

Brandon Fox

Numerade Educator

01:30

Problem 22

Solve each system of equations. If the system has no solution, state that it is inconsistent. Graph the lines of the system.
$$
\left\{\begin{array}{r}
x+3 y=5 \\
2 x-3 y=-8
\end{array}\right.
$$

Brandon Fox

Numerade Educator

01:05

Problem 23

Solve each system of equations. If the system has no solution, state that it is inconsistent. Graph the lines of the system.
$$
\left\{\begin{array}{r}
3 x=24 \\
x+2 y=0
\end{array}\right.
$$

Brandon Fox

Numerade Educator

01:33

Problem 24

Solve each system of equations. If the system has no solution, state that it is inconsistent. Graph the lines of the system.
$$
\left\{\begin{array}{r}
4 x+5 y=-3 \\
-2 y=-8
\end{array}\right.
$$

Brandon Fox

Numerade Educator

01:59

Problem 25

Solve each system of equations. If the system has no solution, state that it is inconsistent. Graph the lines of the system.
$$
\left\{\begin{array}{l}
3 x-6 y=2 \\
5 x+4 y=1
\end{array}\right.
$$

Brandon Fox

Numerade Educator

01:55

Problem 26

Solve each system of equations. If the system has no solution, state that it is inconsistent. Graph the lines of the system.
$$
\left\{\begin{array}{l}
2 x+4 y=\frac{2}{3} \\
3 x-5 y=-10
\end{array}\right.
$$

Brandon Fox

Numerade Educator

01:09

Problem 27

Solve each system of equations. If the system has no solution, state that it is inconsistent. Graph the lines of the system.
$$
\left\{\begin{array}{l}
2 x+y=1 \\
4 x+2 y=3
\end{array}\right.
$$

Brandon Fox

Numerade Educator

01:06

Problem 28

Solve each system of equations. If the system has no solution, state that it is inconsistent. Graph the lines of the system.
$$
\left\{\begin{array}{r}
x-y=5 \\
-3 x+3 y=2
\end{array}\right.
$$

Brandon Fox

Numerade Educator

01:12

Problem 29

Solve each system of equations. If the system has no solution, state that it is inconsistent. Graph the lines of the system.
$$
\left\{\begin{array}{l}
2 x-y=0 \\
4 x+2 y=12
\end{array}\right.
$$

Brandon Fox

Numerade Educator

01:29

Problem 30

Solve each system of equations. If the system has no solution, state that it is inconsistent. Graph the lines of the system.
$$
\left\{\begin{array}{l}
3 x+3 y=-1 \\
4 x+y=\frac{8}{3}
\end{array}\right.
$$

Brandon Fox

Numerade Educator

00:52

Problem 31

Solve each system of equations. If the system has no solution, state that it is inconsistent.
$$
\left\{\begin{array}{r}
x+2 y=4 \\
2 x+4 y=8
\end{array}\right.
$$

Brandon Fox

Numerade Educator

01:22

Problem 32

Solve each system of equations. If the system has no solution, state that it is inconsistent.
$$
\left\{\begin{array}{l}
3 x-y=7 \\
9 x-3 y=21
\end{array}\right.
$$

Brandon Fox

Numerade Educator

01:38

Problem 33

Solve each system of equations. If the system has no solution, state that it is inconsistent.
$$
\left\{\begin{array}{r}
2 x-3 y=-1 \\
10 x+y=11
\end{array}\right.
$$

Brandon Fox

Numerade Educator

01:33

Problem 34

Solve each system of equations. If the system has no solution, state that it is inconsistent.
$$
\left\{\begin{array}{l}
3 x-2 y=0 \\
5 x+10 y=4
\end{array}\right.
$$

Brandon Fox

Numerade Educator

01:23

Problem 35

Solve each system of equations. If the system has no solution, state that it is inconsistent.
$$
\left\{\begin{array}{r}
2 x+3 y=6 \\
x-y=\frac{1}{2}
\end{array}\right.
$$

Brandon Fox

Numerade Educator

01:17

Problem 36

Solve each system of equations. If the system has no solution, state that it is inconsistent.
$$
\left\{\begin{aligned}
\frac{1}{2} x+y &=-2 \\
x-2 y &=8
\end{aligned}\right.
$$

Brandon Fox

Numerade Educator

01:32

Problem 37

Solve each system of equations. If the system has no solution, state that it is inconsistent.
$$
\left\{\begin{array}{l}
\frac{1}{2} x+\frac{1}{3} y=3 \\
\frac{1}{4} x-\frac{2}{3} y=-1
\end{array}\right.
$$

Erika Bustos

Numerade Educator

02:36

Problem 38

Solve each system of equations. If the system has no solution, state that it is inconsistent.
$$
\left\{\begin{array}{l}
\frac{1}{3} x-\frac{3}{2} y=-5 \\
\frac{3}{4} x+\frac{1}{3} y=11
\end{array}\right.
$$

Erika Bustos

Numerade Educator

01:26

Problem 39

Solve each system of equations. If the system has no solution, state that it is inconsistent.
$$
\left\{\begin{array}{l}
3 x-6 y=7 \\
5 x-2 y=5
\end{array}\right.
$$

Brandon Fox

Numerade Educator

01:20

Problem 40

Solve each system of equations. If the system has no solution, state that it is inconsistent.
$$
\left\{\begin{aligned}
2 x-y &=-1 \\
x+\frac{1}{2} y &=\frac{3}{2}
\end{aligned}\right.
$$

Brandon Fox

Numerade Educator

04:18

Problem 41

Solve each system of equations. If the system has no solution, state that it is inconsistent.
$$
\left\{\begin{array}{l}
\frac{1}{x}+\frac{1}{y}=8 \\
\frac{3}{x}-\frac{5}{y}=0
\end{array}\right.
$$

Shakiyla Huggins

Numerade Educator

04:16

Problem 42

Solve each system of equations. If the system has no solution, state that it is inconsistent.
$$
\left\{\begin{array}{l}
\frac{4}{x}-\frac{3}{y}=0 \\
\frac{6}{x}+\frac{3}{2 y}=2
\end{array}\right.
$$

Shakiyla Huggins

Numerade Educator

02:31

Problem 43

Solve each system of equations. If the system has no solution, state that it is inconsistent.
$$
\left\{\begin{aligned}
x+y &=9 \\
2 x &-z=13 \\
3 y+2 z &=7
\end{aligned}\right.
$$

Erika Bustos

Numerade Educator

02:33

Problem 44

Solve each system of equations. If the system has no solution, state that it is inconsistent.
$$
\left\{\begin{array}{l}
2 x+y =-4 \\
-2 y+4 z =0 \\
3 x -2 z=-11
\end{array}\right.
$$

Erika Bustos

Numerade Educator

02:47

Problem 45

Solve each system of equations. If the system has no solution, state that it is inconsistent.
$$
\left\{\begin{array}{rr}
x-2 y+3 z= & 7 \\
2 x+y+z= & 4 \\
-3 x+2 y-2 z= & -10
\end{array}\right.
$$

Erika Bustos

Numerade Educator

10:30

Problem 46

Solve each system of equations. If the system has no solution, state that it is inconsistent.
$$
\left\{\begin{array}{rr}
2 x+y-3 z= & 0 \\
-2 x+2 y+z= & -7 \\
3 x-4 y-3 z= & 7
\end{array}\right.
$$

Shakiyla Huggins

Numerade Educator

02:56

Problem 47

Solve each system of equations. If the system has no solution, state that it is inconsistent.
$$
\left\{\begin{aligned}
x-y-z &=1 \\
2 x+3 y+z &=2 \\
3 x+2 y &=0
\end{aligned}\right.
$$

Shakiyla Huggins

Numerade Educator

03:38

Problem 48

Solve each system of equations. If the system has no solution, state that it is inconsistent.
$$
\left\{\begin{array}{r}
2 x-3 y-z=0 \\
-x+2 y+z=5 \\
3 x-4 y-z=1
\end{array}\right.
$$

Shakiyla Huggins

Numerade Educator

07:01

Problem 49

Solve each system of equations. If the system has no solution, state that it is inconsistent.
$$
\left\{\begin{array}{rr}
x-y-z= & 1 \\
-x+2 y-3 z= & -4 \\
3 x-2 y-7 z= & 0
\end{array}\right.
$$

Shaikh Sadi

Numerade Educator

09:05

Problem 50

Solve each system of equations. If the system has no solution, state that it is inconsistent.
$$
\left\{\begin{array}{r}
2 x-3 y-z=0 \\
3 x+2 y+2 z=2 \\
x+5 y+3 z=2
\end{array}\right.
$$

Shaikh Sadi

Numerade Educator

03:53

Problem 51

Solve each system of equations. If the system has no solution, state that it is inconsistent.
$$
\left\{\begin{array}{r}
2 x-2 y+3 z=6 \\
4 x-3 y+2 z=0 \\
-2 x+3 y-7 z=1
\end{array}\right.
$$

David Mccaslin

Numerade Educator

03:31

Problem 52

Solve each system of equations. If the system has no solution, state that it is inconsistent.
$$
\left\{\begin{array}{l}
3 x-2 y+2 z=6 \\
7 x-3 y+2 z=-1 \\
2 x-3 y+4 z=0
\end{array}\right.
$$

David Mccaslin

Numerade Educator

04:43

Problem 53

Solve each system of equations. If the system has no solution, state that it is inconsistent.
$$
\left\{\begin{array}{r}
x+y-z=6 \\
3 x-2 y+z=-5 \\
x+3 y-2 z=14
\end{array}\right.
$$

David Mccaslin

Numerade Educator

04:55

Problem 54

Solve each system of equations. If the system has no solution, state that it is inconsistent.
$$
\left\{\begin{array}{r}
x-y+z=-4 \\
2 x-3 y+4 z=-15 \\
5 x+y-2 z=12
\end{array}\right.
$$

David Mccaslin

Numerade Educator

05:08

Problem 55

Solve each system of equations. If the system has no solution, state that it is inconsistent.
$$
\left\{\begin{array}{r}
x+2 y-z=-3 \\
2 x-4 y+z=-7 \\
-2 x+2 y-3 z=4
\end{array}\right.
$$

David Mccaslin

Numerade Educator

06:55

Problem 56

Solve each system of equations. If the system has no solution, state that it is inconsistent.
$$
\left\{\begin{array}{rr}
x+4 y-3 z= & -8 \\
3 x-y+3 z= & 12 \\
x+y+6 z= & 1
\end{array}\right.
$$

David Mccaslin

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

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

Numerade Educator

02:11

Problem 59

Orbital Launches In 2017 there was a total of 469 commercial and noncommercial orbital launches worldwide. In addition, the number of noncommercial orbital launches was 31 more than half the number of commercial orbital launches. Determine the number of commercial and noncommercial orbital launches in $2017 .$

Erika Bustos

Numerade Educator

02:02

Problem 60

Movie Theater Tickets A movie theater charges $\$ 9.00$ for adults and $\$ 7.00$ 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?

Erika Bustos

Numerade Educator

04:10

Problem 61

Mixing Nuts 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?

David Mccaslin

Numerade Educator

04:40

Problem 62

Mixing a Solution 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?

David Mccaslin

Numerade Educator

04:50

Problem 63

Presale Order 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 $\$ 486,000 .$ If the price of a smartphone and tablet together is $\$ 1665,$ how much does each device cost?

David Mccaslin

Numerade Educator

07:56

Problem 64

Financial Planning A recently retired couple needs $\$ 12,000$ per year to supplement their Social Security. They have $\$ 300,000$ to invest to obtain this income. They have decided on two investment options: AA bonds yielding $5 \%$ per annum and a Bank Certificate yielding $2.5 \%$.
(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?

Peter Lynch

Numerade Educator

02:35

Problem 65

Computing Wind Speed 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.

Erika Bustos

Numerade Educator

03:07

Problem 66

Computing Wind Speed 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?

David Mccaslin

Numerade Educator

02:15

Problem 67

Restaurant Management 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?

Erika Bustos

Numerade Educator

02:36

Problem 68

Cost of Fast Food 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?

Jerelyn Nevil

Numerade Educator

03:34

Problem 69

Computing a Refund 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?

Erika Bustos

Numerade Educator

01:45

Problem 70

Finding the Current of a Stream 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.

Erika Bustos

Numerade Educator

02:32

Problem 71

Pharmacy 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 $\mathrm{D} .$ How many milligrams of each compound should be mixed to fill the prescription?

Dharmendra Jain

Numerade Educator

01:49

Problem 72

Pharmacy 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

Numerade Educator

02:56

Problem 73

Curve Fitting 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)

Erika Bustos

Numerade Educator

03:50

Problem 74

Curve Fitting 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)

Erika Bustos

Numerade Educator

02:17

Problem 75

IS-LM Model in Economics 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.

Erika Bustos

Numerade Educator

01:43

Problem 76

IS-LM Model in Economics 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.

Erika Bustos

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 $I_{3}$

Dharmendra Jain

Numerade Educator

07:50

Problem 78

Electricity: Kirchhoff's 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 $I_{3}$

David Mccaslin

Numerade Educator

07:07

Problem 79

Theater Revenues 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?

Erika Bustos

Numerade Educator

04:26

Problem 80

Theater Revenues 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?

Erika Bustos

Numerade Educator

12:22

Problem 81

Nutrition 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 $3 \mathrm{~g}$ of protein, $16 \mathrm{~g}$ of carbohydrates, and $10 \mathrm{mg}$ of calcium. Each glass of $2 \%$ milk has $9 \mathrm{~g}$ of protein, $13 \mathrm{~g}$ of carbohydrates, and $300 \mathrm{mg}$ of calcium. How many servings of each food should the dietitian provide for the patient?

David Mccaslin

Numerade Educator

03:02

Problem 82

Investments 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

Numerade Educator

07:05

Problem 83

Prices of Fast Food 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 $\$ 0.90 .$

David Mccaslin

Numerade Educator

08:24

Problem 84

Prices of Fast Food 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.

David Mccaslin

Numerade Educator

08:32

Problem 85

Painting a House Three painters (Beth, Dan, and Edie), working together, can paint the exterior of a home in 10 hours (h). Dan and Edie together have painted a similar house in $15 \mathrm{~h}$. One day, all three worked on this same kind of house for $4 \mathrm{~h},$ after which Edie left. Beth and Dan 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?

David Mccaslin

Numerade Educator

02:44

Problem 86

Challenge Problem Solve for $x$ and $y,$ assuming $a \neq 0$ and $b \neq 0$
$$
\left\{\begin{array}{l}
a x+b y=a+b \\
a b x-b^{2} y=b^{2}-a b
\end{array}\right.
$$

Ziya Ogron

Numerade Educator

14:40

Problem 87

Challenge Problem Solve for $x, y,$ and $z,$ assuming $a \neq 0, b \neq 0,$ and $c \neq 0$
$$
\left\{\begin{array}{l}
a x+b y+c z =a+b+c \\
a^{2} x+b^{2} y+c^{2} z =a c+a b+b c \\
a b x+b c y \quad \quad=b c+a c
\end{array}\right.
$$

David Mccaslin

Numerade Educator

02:27

Problem 88

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

Numerade Educator

02:15

Problem 89

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

Dharmendra Jain

Numerade Educator

02:09

Problem 90

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

Numerade Educator

02:19

Problem 91

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
Graph $f(x)=-3^{1-x}+2$

David Mccaslin

Numerade Educator

07:49

Problem 92

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
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) $\frac{1}{2}(3 x-5)^{-\frac{1}{2}} \cdot 3 \cdot(x+3)^{-\frac{1}{2}}-\frac{1}{2}(x+3)^{-\frac{3}{2}}(3 x-5)^{\frac{1}{2}}$

Stephanie Carter

Numerade Educator

03:27

Problem 93

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
Find the exact value of $\sin ^{-1}\left[\sin \left(-\frac{10 \pi}{9}\right)\right] .$

Stephanie Carter

Numerade Educator

04:52

Problem 94

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
Write $-\sqrt{3}+i$ in polar form and in exponential form.

Stephanie Carter

Numerade Educator

02:10

Problem 95

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
If $A=\{2,4,6, \ldots, 30\} \quad$ and $\quad B=\{3,6,9, \ldots, 30\}$ find $A \cap B$

David Mccaslin

Numerade Educator

02:24

Problem 96

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
Find an equation of an ellipse if the center is at the origin, the length of the major axis is 20 along the $x$ -axis, and the length of the minor axis is 12 .

David Mccaslin

Numerade Educator

06:27

Problem 97

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
If $z=6 e^{i \frac{7 \pi}{4}}$ and $w=2 e^{i \frac{5 \pi}{6}},$ find $z w$ and $\frac{z}{w} .$ Write the answers in polar form and in exponential form.

Stephanie Carter

Numerade Educator

03:57

Problem 98

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
Find the principal needed now to get $\$ 5000$ after 18 months at $4 \%$ interest compounded monthly.

Stephanie Carter

Numerade Educator

03:12

Problem 99

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
Find the average rate of change of $f(x)=\cos ^{-1} x$ from $x=-\frac{1}{2}$ to $x=\frac{1}{2}$

Stephanie Carter

Numerade Educator

08:58

Problem 100

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
Find the area of the triangle with vertices at $(0,5),(3,9),$ and (12,0)

Stephanie Carter

Numerade Educator