Algebra for College Students

Problem 9

Solve each system by graphing.
$$\begin{aligned}&y=2 x-1\\&2 y=x-2 \end{aligned}$$

Amy Jiang

Numerade Educator

Problem 10

Solve each system by graphing.
$$\begin{aligned}
&y=2 x+1\\
&x+y=-2
\end{aligned}$$

Amy Jiang

Numerade Educator

Problem 11

Solve each system by graphing.
$$\begin{aligned}
&y=x-3\\
&x-2 y=4
\end{aligned}$$

Amy Jiang

Numerade Educator

Problem 12

Solve each system by graphing.
$$\begin{aligned}&y=-3 x\\&x+y=2\end{aligned}$$

Amy Jiang

Numerade Educator

00:49

Problem 13

Solve each system by graphing.
$$\begin{aligned}&y=2 x+4\\&3 x+y=-1\end{aligned}$$

Amy Jiang

Numerade Educator

01:07

Problem 14

Solve each system by graphing.
$$\begin{aligned}&3 x-2 y=6\\&3 x+2 y=6\end{aligned}$$

Amy Jiang

Numerade Educator

01:05

Problem 15

Solve each system by graphing.
$$\begin{aligned}&y=-\frac{1}{2} x+4\\&x+2 y=8\end{aligned}$$

Amy Jiang

Numerade Educator

00:54

Problem 16

Solve each system by graphing.
$$\begin{aligned}&2 x-3 y=6\\&y=\frac{2}{3} x-2\end{aligned}$$

Amy Jiang

Numerade Educator

00:24

Problem 17

Solve each system by graphing.
$$\begin{aligned}&2 y-2 x=2\\&2 y-2 x=6\end{aligned}$$

Amy Jiang

Numerade Educator

00:38

Problem 18

Solve each system by graphing.
$$\begin{aligned}&3 y-3 x=9\\&x-y=1\end{aligned}$$

Amy Jiang

Numerade Educator

00:27

Problem 19

Solve each system by graphing.
$$\begin{array}{l}y=-\frac{1}{4} x \\x+4 y=8\end{array}$$

Amy Jiang

Numerade Educator

00:28

Problem 20

Solve each system by graphing.
$$\begin{array}{l}y=-\frac{2}{3} x \\2 x+3 y=5\end{array}$$

Amy Jiang

Numerade Educator

01:28

Problem 21

The graphs of the following systems are given in (a) through (d). Match each system with the correct graph.
$$\begin{array}{l}5 x+4 y=7 \\x-3 y=9\end{array}$$

Amy Jiang

Numerade Educator

01:54

Problem 22

The graphs of the following systems are given in (a) through (d). Match each system with the correct graph.
$$\begin{aligned}&3 x-5 y=-9\\&5 x-6 y=-8\end{aligned}$$

Amy Jiang

Numerade Educator

01:09

Problem 23

The graphs of the following systems are given in (a) through (d). Match each system with the correct graph.
$$\begin{aligned}&4 x-5 y=-2\\&3 y-x=-3\end{aligned}$$

Amy Jiang

Numerade Educator

01:11

Problem 24

The graphs of the following systems are given in (a) through (d). Match each system with the correct graph.
$$\begin{aligned}&4 x+5 y=-2\\&4 y-x=11\end{aligned}$$

Amy Jiang

Numerade Educator

00:53

Graph each pair of equations on a graphing calculator using a window that shows the point of intersection. Use the intersect feature to find the solution.
GRAPH CANNOT COPY.
$$\begin{aligned}&y=4 x-22.35\\&y=-6 x+50.15\end{aligned}$$

Amy Jiang

Numerade Educator

01:03

Problem 26

Amy Jiang

Numerade Educator

00:41

Problem 27

Amy Jiang

Numerade Educator

Problem 28

Amy Jiang

Numerade Educator

01:11

Problem 29

Amy Jiang

Numerade Educator

01:10

Problem 30

Amy Jiang

Numerade Educator

01:49

Problem 31

Amy Jiang

Numerade Educator

Problem 32

Amy Jiang

Numerade Educator

01:05

Problem 33

Amy Jiang

Numerade Educator

00:49

Problem 34

Amy Jiang

Numerade Educator

00:47

Problem 35

Solve each system by substitution. Determine whether the equations are independent, dependent, or inconsistent.
$$\begin{aligned}&y=4 x-1\\&y=x+8\end{aligned}$$

Amy Jiang

Numerade Educator

00:47

Problem 36

Solve each system by substitution. Determine whether the equations are independent, dependent, or inconsistent.
$$\begin{aligned}&y=-3 x+19\\&y=2 x-1\end{aligned}$$

Amy Jiang

Numerade Educator

00:42

Problem 37

Solve each system by substitution. Determine whether the equations are independent, dependent, or inconsistent.
$$\begin{aligned}&y=-2 x\\&y=4 x+12\end{aligned}$$

Amy Jiang

Numerade Educator

Problem 38

Solve each system by substitution. Determine whether the equations are independent, dependent, or inconsistent.
$$\begin{aligned}&y=-4 x-7\\&y=3 x\end{aligned}$$

Amy Jiang

Numerade Educator

01:03

Problem 39

Solve each system by substitution. Determine whether the equations are independent, dependent, or inconsistent.
$$\begin{aligned}&y=\frac{1}{3} x+2\\&y=-\frac{1}{2} x+7\end{aligned}$$

Amy Jiang

Numerade Educator

01:11

Problem 40

Solve each system by substitution. Determine whether the equations are independent, dependent, or inconsistent.
$$\begin{array}{l}y=-\frac{2}{5} x-2 \\y=\frac{3}{2} x+17\end{array}$$

Amy Jiang

Numerade Educator

Problem 41

Solve each system by substitution. Determine whether the equations are independent, dependent, or inconsistent.
$$\begin{aligned}&y=x-5\\&2 x-5 y=1\end{aligned}$$

Amy Jiang

Numerade Educator

00:52

Problem 42

Solve each system by substitution. Determine whether the equations are independent, dependent, or inconsistent.
$$\begin{aligned}&y=x+4\\&3 y-5 x=6\end{aligned}$$

Amy Jiang

Numerade Educator

Problem 43

Solve each system by substitution. Determine whether the equations are independent, dependent, or inconsistent.
$$\begin{aligned}&x=2 y-7\\&3 x+2 y=-5\end{aligned}$$

Amy Jiang

Numerade Educator

00:45

Problem 44

Solve each system by substitution. Determine whether the equations are independent, dependent, or inconsistent.
$$\begin{aligned}&x=y+3\\&3 x-2 y=4\end{aligned}$$

Amy Jiang

Numerade Educator

01:06

Problem 45

Solve each system by substitution. Determine whether the equations are independent, dependent, or inconsistent.
$$\begin{aligned}&y=2 x-30\\&\frac{1}{5} x-\frac{1}{2} y=-1\end{aligned}$$

Amy Jiang

Numerade Educator

01:12

Problem 46

Solve each system by substitution. Determine whether the equations are independent, dependent, or inconsistent.
$$\begin{aligned}&3 x-5 y=4\\&y=\frac{3}{4} x-2\end{aligned}$$

Amy Jiang

Numerade Educator

01:02

Problem 47

Solve each system by substitution. Determine whether the equations are independent, dependent, or inconsistent.
$$\begin{aligned}&2 x+y=9\\&2 x-5 y=15\end{aligned}$$

Amy Jiang

Numerade Educator

Problem 48

Solve each system by substitution. Determine whether the equations are independent, dependent, or inconsistent.
$$\begin{aligned}&3 y-x=0\\&x-4 y=-2\end{aligned}$$

Amy Jiang

Numerade Educator

00:33

Problem 49

Solve each system by substitution. Determine whether the equations are independent, dependent, or inconsistent.
$$\begin{aligned}&x-y=0\\&2 x+3 y=35\end{aligned}$$

Amy Jiang

Numerade Educator

00:43

Problem 50

Solve each system by substitution. Determine whether the equations are independent, dependent, or inconsistent.
$$\begin{aligned}&2 y=x+6\\&-3 x+2 y=-2\end{aligned}$$

Amy Jiang

Numerade Educator

Problem 51

Solve each system by substitution. Determine whether the equations are independent, dependent, or inconsistent.
$$\begin{aligned}&x+y=40\\&0.2 x+0.8 y=23\end{aligned}$$

Amy Jiang

Numerade Educator

00:54

Problem 52

Solve each system by substitution. Determine whether the equations are independent, dependent, or inconsistent.
$$\begin{aligned}&x-y=10\\&0.1 x+0.5 y=13\end{aligned}$$

Amy Jiang

Numerade Educator

00:36

Problem 53

Solve each system by substitution. Determine whether the equations are independent, dependent, or inconsistent.
$$\begin{aligned}&y=2 x-5\\&y+1=2(x-2)\end{aligned}$$

Amy Jiang

Numerade Educator

00:53

Problem 54

Solve each system by substitution. Determine whether the equations are independent, dependent, or inconsistent.
$$\begin{aligned}&2 x-y=3\\&2 y=4 x-6\end{aligned}$$

Amy Jiang

Numerade Educator

00:46

Problem 55

Solve each system by substitution. Determine whether the equations are independent, dependent, or inconsistent.
$$\begin{aligned}&x-y=5\\&2 x=2 y+14\end{aligned}$$

Amy Jiang

Numerade Educator

00:33

Problem 56

Solve each system by substitution. Determine whether the equations are independent, dependent, or inconsistent.
$$\begin{array}{l}2 x-y=4 \\2 x-y=3\end{array}$$

Amy Jiang

Numerade Educator

01:20

Problem 57

Solve each system by substitution. Determine whether the equations are independent, dependent, or inconsistent.
$$\begin{aligned}&y=\frac{5}{7} x\\&x=-\frac{2}{3} y\end{aligned}$$

Amy Jiang

Numerade Educator

01:02

Problem 58

Solve each system by substitution. Determine whether the equations are independent, dependent, or inconsistent.
$$\begin{aligned}&7 y=9 x\\&-3 x=4 y\end{aligned}$$

Amy Jiang

Numerade Educator

Problem 59

Solve each system by substitution. Determine whether the equations are independent, dependent, or inconsistent.
$$\begin{aligned}&3(y-1)=2(x-3)\\&3 y-2 x=-3\end{aligned}$$

Amy Jiang

Numerade Educator

Problem 60

Solve each system by substitution. Determine whether the equations are independent, dependent, or inconsistent.
$$\begin{aligned}&y=3(x-4)\\&3 x-y=12\end{aligned}$$

Amy Jiang

Numerade Educator

00:20

Problem 61

Solve each system by substitution. Determine whether the equations are independent, dependent, or inconsistent.
$$\begin{aligned}&y=3 x\\&y=3 x+1\end{aligned}$$

Amy Jiang

Numerade Educator

00:25

Problem 62

Solve each system by substitution. Determine whether the equations are independent, dependent, or inconsistent.
$$\begin{aligned}&y=3 x-4\\&y=3 x+4\end{aligned}$$

Amy Jiang

Numerade Educator

01:02

Problem 63

Solve each system by the substitution method.
$$\begin{aligned}&y=\frac{5}{2} x\\&x+3 y=3\end{aligned}$$

Amy Jiang

Numerade Educator

01:52

Problem 64

Solve each system by the substitution method.
$$\begin{aligned}&6 x-3 y=3\\&10 x=y+7\end{aligned}$$

Amy Jiang

Numerade Educator

00:46

Problem 65

Solve each system by the substitution method.
$$\begin{aligned}&x+y=4\\&x-y=5\end{aligned}$$

Amy Jiang

Numerade Educator

01:24

Problem 66

Solve each system by the substitution method.
$$\begin{aligned}&3 x-6 y=5\\&2 y=4 x-6\end{aligned}$$

Amy Jiang

Numerade Educator

Problem 67

Solve each system by the substitution method.
$$\begin{aligned}&2 x-4 y=0\\&6 x+8 y=5\end{aligned}$$

Amy Jiang

Numerade Educator

01:42

Problem 68

Solve each system by the substitution method.
$$\begin{array}{r}-3 x+10 y=4 \\6 x-5 y=1\end{array}$$

Amy Jiang

Numerade Educator

01:22

Problem 69

Solve each system by the substitution method.
$$\begin{array}{l}3 x+y=2 \\-x-3 y=6\end{array}$$

Amy Jiang

Numerade Educator

00:51

Problem 70

Solve each system by the substitution method.
$$\begin{array}{r}x+3 y=2 \\-x+y=1\end{array}$$

Amy Jiang

Numerade Educator

02:16

Problem 71

Solve each system by the substitution method.
$$\begin{aligned}&-9 x+6 y=3\\&18 x+30 y=1\end{aligned}$$

Amy Jiang

Numerade Educator

01:34

Problem 72

Solve each system by the substitution method.
$$\begin{aligned}x+6 y &=-2 \\5 x-20 y &=5\end{aligned}$$

Amy Jiang

Numerade Educator

00:37

Problem 73

Solve each system by the substitution method.
$$\begin{aligned}&y=-2 x\\&3 y-x=1\end{aligned}$$

Amy Jiang

Numerade Educator

00:37

Problem 74

Solve each system by the substitution method.
$$\begin{aligned}&y=2 x\\&15 x-10 y=-2\end{aligned}$$

Amy Jiang

Numerade Educator

01:06

Problem 75

Solve each system by the substitution method.
$$\begin{aligned}x &=-6 y+1 \\2 y &=-5 x\end{aligned}$$

Amy Jiang

Numerade Educator

01:26

Problem 76

Solve each system by the substitution method.
$$\begin{aligned}x &=-3 y+2 \\7 y &=3 x\end{aligned}$$

Amy Jiang

Numerade Educator

00:55

Problem 77

Solve each system by the substitution method.
$$\begin{array}{c}x-y=0.1 \\2 x-3 y=-0.5\end{array}$$

Amy Jiang

Numerade Educator

01:14

Problem 78

Solve each system by the substitution method.
$$\begin{array}{l}y-2 x=-7.5 \\3 x-5 y=3.2\end{array}$$

Amy Jiang

Numerade Educator

01:36

Problem 79

Write a system of two equations in two unknowns for each problem. Solve each system by substitution.
Rectangular patio. The length of a rectangular patio is 12 feet greater than the width. If the perimeter is 84 feet, then what are the length and width?

Amy Jiang

Numerade Educator

01:19

Problem 80

Write a system of two equations in two unknowns for each problem. Solve each system by substitution.
Rectangular notepad. The length of a rectangular notepad is $2 \mathrm{cm}$ longer than twice the width. If the perimeter is $34 \mathrm{cm},$ then what are the length and width?

Amy Jiang

Numerade Educator

01:07

Problem 81

Write a system of two equations in two unknowns for each problem. Solve each system by substitution.
Rectangular table. The width of a rectangular table is $1 \mathrm{ft}$ less than half of the length. If the perimeter is $28 \mathrm{ft}$, then what are the length and width?

Amy Jiang

Numerade Educator

Problem 82

Write a system of two equations in two unknowns for each problem. Solve each system by substitution.
Rectangular painting. The width of a rectangular painting is two-thirds of its length. If the perimeter is 60 in., then what are the length and width?

Amy Jiang

Numerade Educator

01:05

Problem 83

Write a system of two equations in two unknowns for each problem. Solve each system by substitution.
Sum and difference. The sum of two numbers is 10 and their difference is $3 .$ Find the numbers.

Amy Jiang

Numerade Educator

01:10

Problem 84

Write a system of two equations in two unknowns for each problem. Solve each system by substitution.
Sum and difference. The sum of two numbers is 51 and their difference is $26 .$ Find the numbers.

Amy Jiang

Numerade Educator

00:56

Problem 85

Write a system of two equations in two unknowns for each problem. Solve each system by substitution.
Sum and difference. The sum of two numbers is 1 and their difference is $20 .$ Find the numbers.

Amy Jiang

Numerade Educator

Problem 86

Write a system of two equations in two unknowns for each problem. Solve each system by substitution.
Sum and difference. The sum of two numbers is 5 and their difference is $30 .$ Find the numbers.

Amy Jiang

Numerade Educator

Problem 87

Write a system of two equations in two unknowns for each problem. Solve each system by substitution.
Flying to Vegas. Two hundred people were on a charter flight to Las Vegas. Some paid $\$ 200$ for their tickets and some paid $\$ 250 .$ If the total revenue for the flight was $\$ 44,000$ then how many tickets of each type were sold?

Amy Jiang

Numerade Educator

01:13

Problem 88

Write a system of two equations in two unknowns for each problem. Solve each system by substitution.
Annual concert. A total of 150 tickets were sold for the annual concert to students and nonstudents. Student tickets were $\$ 5$ and nonstudent tickets were $\$ 8 .$ If the total revenue for the concert was $\$ 930,$ then how many tickets of each type were sold?

Amy Jiang

Numerade Educator

01:31

Problem 89

Write a system of two equations in two unknowns for each problem. Solve each system by substitution.
Annual play. There were twice as many tickets sold to nonstudents than to students for the annual play. Student tickets were $\$ 6$ and nonstudent tickets were $\$ 11 .$ If the total revenue for the play was $\$ 1540,$ then how many tickets of each type were sold?

Amy Jiang

Numerade Educator

01:55

Problem 90

Write a system of two equations in two unknowns for each problem. Solve each system by substitution.
Soccer game. There were 1000 more students at the soccer game than nonstudents. Student tickets were $\$ 8.50$ and monstudent tickets were $\$ 13.25 .$ If the total revenue for the game was $\$ 75,925,$ then how many tickets of each type were sold?

Amy Jiang

Numerade Educator

01:56

Problem 91

Write a system of two equations in two unknowns for each problem. Solve each system by substitution.
Mixing investments. Helen invested $\$ 40,000$ and received a total of $\$ 2300$ in interest after one year. If part of the money returned $5 \%$ and the remainder $8 \%,$ then how much did she invest at each rate?

Amy Jiang

Numerade Educator

01:55

Problem 92

Write a system of two equations in two unknowns for each problem. Solve each system by substitution.
Investing her bonus. Donna invested her $\$ 33,000$ bonus and received a total of $\$ 970$ in interest after one year. If part of the money returned $4 \%$ and the remainder $2.25 \%$ then how much did she invest at each rate?

Amy Jiang

Numerade Educator

02:27

Problem 93

Write a system of two equations in two unknowns for each problem. Solve each system by substitution.
Mixing acid. A chemist wants to mix a $5 \%$ acid solution with a $25 \%$ acid solution to obtain 50 liters of a $20 \%$ acid solution. How many liters of each solution should be used?

Daniel Carr

Numerade Educator

01:48

Problem 94

Write a system of two equations in two unknowns for each problem. Solve each system by substitution.
Mixing fertilizer. A farmer wants to mix a liquid fertilizer that contains $2 \%$ nitrogen with one that contains $10 \%$ nitrogen to obtain 40 gallons of a fertilizer that contains $8 \%$ nitrogen. How many gallons of each fertilizer should be used?

Ashley Volpe

Numerade Educator

01:56

Problem 95

Write a system of two equations in two unknowns for each problem. Solve each system by substitution.
Different interest rates. Mrs. Brighton invested $\$ 30,000$ and received a total of $\$ 2300$ in interest. If she invested part of the money at $10 \%$ and the remainder at $5 \%,$ then how much did she invest at each rate?

Amy Jiang

Numerade Educator

06:15

Problem 96

Write a system of two equations in two unknowns for each problem. Solve each system by substitution.
Different growth rates. The combined population of Marysville and Springfield was $25,000$ in $2000 .$ By 2005 the population of Marysville had increased by $10 \%,$ while Springfield had increased by $9 \%$. If the total population increased by 2380 people, then what was the population of each city in $2000 ?$

Wesley Hines

Numerade Educator

00:41

Problem 97

Write a system of two equations in two unknowns for each problem. Solve each system by substitution.
Finding numbers. The sum of two numbers is $2,$ and their difference is $26 .$ Find the numbers.

Amy Jiang

Numerade Educator

00:37

Problem 98

Write a system of two equations in two unknowns for each problem. Solve each system by substitution.
Finding more numbers. The sum of two numbers is $-16,$ and their difference is $8 .$ Find the numbers.

Amy Jiang

Numerade Educator

01:48

Problem 99

Write a system of two equations in two unknowns for each problem. Solve each system by substitution.
Toasters and vacations. During one week a land developer gave away Florida vacation coupons or toasters to 100 potential customers who listened to a sales presentation. It costs the developer $\$ 6$ for a toaster and $\$ 24$ for a Florida vacation coupon. If his bill for prizes that week was $\$ 708,$ then how many of each prize did he give away?

Kenny Mesadieu

Numerade Educator

01:13

Problem 100

Write a system of two equations in two unknowns for each problem. Solve each system by substitution.
Ticket sales. Tickets for a concert were sold to adults for $\$ 3$ and to students for $\$ 2 .$ If the total receipts were $\$ 824$ and twice as many adult tickets as student tickets were sold, then how many of each were sold?

Amy Jiang

Numerade Educator

02:29

Problem 101

Write a system of two equations in two unknowns for each problem. Solve each system by substitution.
Corporate taxes. According to Bruce Harrell, CPA, the amount of federal income tax for a class C corporation is deductible on the Louisiana state tax return, and the amount of state income tax for a class C corporation is deductible on the federal tax return. So for a state tax rate of $5 \%$ and a federal tax rate of $30 \%,$ we have state tax $=0.05(\text { taxable income }-\text { federal tax })$ and federal tax $=0.30$ (taxable income - state tax). Find the amounts of state and federal income taxes for a class C corporation that has a taxable income of $\$ 100,000.$

Joshua Eastwood

Numerade Educator

03:14

Problem 102

Write a system of two equations in two unknowns for each problem. Solve each system by substitution.
More taxes. Use the information given in Exercise 101 to find the amounts of state and federal income taxes for a class C corporation that has a taxable income of $\$ 300,000$. Use a state tax rate of $6 \%$ and a federal tax rate of $40 \% .$

Ian Maurer

Numerade Educator

Problem 103

Write a system of two equations in two unknowns for each problem. Solve each system by substitution.
Cost accounting. The problems presented in this exercise and the next are encountered in cost accounting. A company has agreed to distribute $20 \%$ of its net income $N$ to its employees as a bonus; $B=0.20 N .$ If the company has income of $\$ 120,000$ before the bonus, the bonus $B$ is deducted from the $\$ 120,000$ as an expense to determine net income; $N=120,000-B .$ Solve the system of two equations in $N$ and $B$ to find the amount of the bonus.

Amy Jiang

Numerade Educator

Problem 104

Write a system of two equations in two unknowns for each problem. Solve each system by substitution.
Bonus and taxes. A company has an income of $\$ 100,000$ before paying taxes and a bonus. The bonus $B$ is to be $20 \%$ of the income after deducting income taxes $T$ but before deducting the bonus. So
$$B=0.20(100,000-T)$$
Because the bonus is a deductible expense, the amount of income tax $T$ at a $40 \%$ rate is $40 \%$ of the income after deducting the bonus. So
$$T=0.40(100,000-B)$$
a) Use the accompanying graph to estimate the values of $T$ and $B$ that satisfy both equations.
b) Solve the system algebraically to find the bonus and the amount of tax.

FIGURE CANNOT COPY

Amy Jiang

Numerade Educator

Problem 105

Write a system of two equations in two unknowns for each problem. Solve each system by substitution.
Textbook case. The accompanying graph shows the cost of producing textbooks and the revenue from the sale of those textbooks.
FIGURE CANNOT COPY
a) What is the cost of producing $10,000$ textbooks?
b) What is the revenue when $10,000$ textbooks are sold?
c) For what number of textbooks is the cost equal to the revenue?
d) The cost of producing zero textbooks is called the fixed cost. Find the fixed cost.

Amy Jiang

Numerade Educator

02:29

Problem 106

Write a system of two equations in two unknowns for each problem. Solve each system by substitution.
Free market. The function $S=5000+200 x$ and $D=9500-100 x$ express the supply $S$ and the demand $D,$ respectively, for a popular compact disc brand as a function of its price $x$ (in dollars).
a) Graph the functions on the same coordinate system.
b) What happens to the supply as the price increases?
c) What happens to the demand as the price increases?
d) The price at which supply and demand are equal is called the equilibrium price. What is the equilibrium price?

Brandon Fox

Numerade Educator

03:13

Problem 107

Discussion. Which of the following equations is not equivalent to $2 x-3 y=6 ?$
a) $3 y-2 x=6$
b) $y=\frac{2}{3} x-2$
c) $x=\frac{3}{2} y+3$
d) $2(x-5)=3 y-4$

Jodi Folley

Numerade Educator