Chapter 7

Systems of Equations and Inequalities

Educators

MB

Problem 1

Can a linear system of three equations have exactly two solutions? Explain why or why not

Linh V.
Numerade Educator

Problem 2

If a given ordered triple solves the system of equations, is that solution unique? If so, explain why. If not, give an example where it is not unique.

MB
Matt B.
Numerade Educator

Problem 3

If a given ordered triple does not solve the system of equations, is there no solution? If so, explain why. If
not, give an example.

Linh V.
Numerade Educator

Problem 4

Using the method of addition, is there only one way to solve the system?

MB
Matt B.
Numerade Educator

Problem 5

Can you explain whether there can be only one method to solve a linear system of equations? If yes, give an example of such a system of equations. If not, explain why not.

Linh V.
Numerade Educator

Problem 6

For the following exercises, determine whether the ordered triple given is the solution to the system of equations.
$$
\begin{aligned} 2 x-6 y+6 z &=-12 \\ x+4 y+5 z &=-1 \quad \text { and }(0,1,-1) \\-x+2 y+3 z &=-1 \end{aligned}
$$

MB
Matt B.
Numerade Educator

Problem 7

For the following exercises, determine whether the ordered triple given is the solution to the system of equations.
$$
\begin{aligned} 6 x-y+3 z &=6 \\ 3 x+5 y+2 z &=0 \quad \text { and }(3,-3,-5) \\ x+y &=0 \end{aligned}
$$

Rachael K.
Numerade Educator

Problem 8

For the following exercises, determine whether the ordered triple given is the solution to the system of equations.
$$
\begin{aligned} 6 x-7 y+z &=2 \\-x-y+3 z &=4 \quad \text { and }(4,2,-6) \\ 2 x+y-z &=1 \end{aligned}
$$

MB
Matt B.
Numerade Educator

Problem 9

For the following exercises, determine whether the ordered triple given is the solution to the system of equations.
$$
\begin{aligned} x-y &=0 \\ x-z &=5 \quad \text { and }(4,4,-1) \\ x-y+z &=-1 \end{aligned}
$$

Check back soon!

Problem 10

For the following exercises, determine whether the ordered triple given is the solution to the system of equations.
$$
\begin{array}{l}{-x-y+2 z=3} \\ {5 x+8 y-3 z=4 \quad \text { and }(4,1,-7)} \\ {-x+3 y-5 z=-5}\end{array}
$$

MB
Matt B.
Numerade Educator

Problem 11

For the following exercises, solve each system by substitution.
$$
\begin{aligned} 3 x-4 y+2 z &=-15 \\ 2 x+4 y+z &=16 \\ 2 x+3 y+5 z &=20 \end{aligned}
$$

Linh V.
Numerade Educator

Problem 12

For the following exercises, solve each system by substitution.
$$
\begin{aligned} 5 x-2 y+3 z &=20 \\ 2 x-4 y-3 z &=-9 \\ x+6 y-8 z &=21 \end{aligned}
$$

MB
Matt B.
Numerade Educator

Problem 13

For the following exercises, solve each system by substitution.
$$
\begin{array}{l}{5 x+2 y+4 z=9} \\ {-3 x+2 y+z=10} \\ {4 x-3 y+5 z=-3}\end{array}
$$

Linh V.
Numerade Educator

Problem 14

For the following exercises, solve each system by substitution.
$$
\begin{aligned} 4 x-3 y+5 z &=31 \\-x+2 y+4 z &=20 \\ x+5 y-2 z &=-29 \end{aligned}
$$

MB
Matt B.
Numerade Educator

Problem 15

For the following exercises, solve each system by substitution.
$$
\begin{aligned} 5 x-2 y+3 z &=4 \\-4 x+6 y-7 z &=-1 \\ 3 x+2 y-z &=4 \end{aligned}
$$

Linh V.
Numerade Educator

Problem 16

For the following exercises, solve each system by substitution.
$$
\begin{aligned} 4 x+6 y+9 z &=0 \\-5 x+2 y-6 z &=3 \\ 7 x-4 y+3 z &=-3 \end{aligned}
$$

MB
Matt B.
Numerade Educator

Problem 17

For the following exercises, solve each system by Gaussian elimination.
$$
\begin{aligned} 2 x-y+3 z &=17 \\-5 x+4 y-2 z &=-46 \\ 2 y+5 z &=-7 \end{aligned}
$$

Linh V.
Numerade Educator

Problem 18

For the following exercises, solve each system by Gaussian elimination.
$$
\begin{aligned} 5 x-6 y+3 z &=50 \\-x+4 y &=10 \\ 2 x-z &=10 \end{aligned}
$$

MB
Matt B.
Numerade Educator

Problem 19

For the following exercises, solve each system by Gaussian elimination.
$$
\begin{aligned} 2 x+3 y-6 z &=1 \\-4 x-6 y+12 z &=-2 \\ x+2 y+5 z &=10 \end{aligned}
$$

Linh V.
Numerade Educator

Problem 20

For the following exercises, solve each system by Gaussian elimination.
$$
\begin{array}{l}{4 x+6 y-2 z=8} \\ {6 x+9 y-3 z=12} \\ {-2 x-3 y+z=-4}\end{array}
$$

MB
Matt B.
Numerade Educator

Problem 21

For the following exercises, solve each system by Gaussian elimination.
$$
\begin{aligned} 2 x+3 y-4 z &=5 \\-3 x+2 y+z &=11 \\-x+5 y+3 z &=4 \end{aligned}
$$

Linh V.
Numerade Educator

Problem 22

For the following exercises, solve each system by Gaussian elimination.
$$
\begin{aligned} 10 x+2 y-14 z &=8 \\-x-2 y-4 z &=-1 \\-12 x-6 y+6 z &=-12 \end{aligned}
$$

MB
Matt B.
Numerade Educator

Problem 23

For the following exercises, solve each system by Gaussian elimination.
$$
\begin{aligned} x+y+z &=14 \\ 2 y+3 z &=-14 \\-16 y-24 z &=-112 \end{aligned}
$$

Linh V.
Numerade Educator

Problem 24

For the following exercises, solve each system by Gaussian elimination.
$$
\begin{aligned} 5 x-3 y+4 z &=-1 \\-4 x+2 y-3 z &=0 \\-x+5 y+7 z &=-11 \end{aligned}
$$

MB
Matt B.
Numerade Educator

Problem 25

For the following exercises, solve each system by Gaussian elimination.
$$
\begin{aligned} x+y+z &=0 \\ 2 x-y+3 z &=0 \\ x-z &=0 \end{aligned}
$$

Linh V.
Numerade Educator

Problem 26

For the following exercises, solve each system by Gaussian elimination.
$$
\begin{aligned} 3 x+2 y-5 z &=6 \\ 5 x-4 y+3 z &=-12 \\ 4 x+5 y-2 z &=15 \end{aligned}
$$

MB
Matt B.
Numerade Educator

Problem 27

For the following exercises, solve each system by Gaussian elimination.
$$
\begin{aligned} x+y+z &=0 \\ 2 x-y+3 z &=0 \\ x-z &=1 \end{aligned}
$$

Linh V.
Numerade Educator

Problem 28

For the following exercises, solve each system by Gaussian elimination.
$$
\begin{aligned} 3 x-\frac{1}{2} y-z &=-\frac{1}{2} \\ 4 x+z &=3 \\-x+\frac{3}{2} y &=\frac{5}{2} \end{aligned}
$$

MB
Matt B.
Numerade Educator

Problem 29

For the following exercises, solve each system by Gaussian elimination.
$$
\begin{aligned} 6 x-5 y+6 z &=38 \\ \frac{1}{5} x-\frac{1}{2} y+\frac{3}{5} z &=1 \\-4 x-\frac{3}{2} y-z &=-74 \end{aligned}
$$

Linh V.
Numerade Educator

Problem 30

For the following exercises, solve each system by Gaussian elimination.
$$
\begin{aligned} \frac{1}{2} x-\frac{1}{5} y+\frac{2}{5} z &=-\frac{13}{10} \\ \frac{1}{4} x-\frac{2}{5} y-\frac{1}{5} z &=-\frac{7}{20} \\-\frac{1}{2} x-\frac{3}{4} y-\frac{1}{2} z &=-\frac{5}{4} \end{aligned}
$$

MB
Matt B.
Numerade Educator

Problem 31

For the following exercises, solve each system by Gaussian elimination.
$$
\begin{array}{l}{-\frac{1}{3} x-\frac{1}{2} y-\frac{1}{4} z=\frac{3}{4}} \\ {-\frac{1}{2} x-\frac{1}{4} y-\frac{1}{2} z=2} \\ {-\frac{1}{4} x-\frac{3}{4} y-\frac{1}{2} z=-\frac{1}{2}}\end{array}
$$

Linh V.
Numerade Educator

Problem 32

For the following exercises, solve each system by Gaussian elimination.
$$
\begin{array}{l}{\frac{1}{2} x-\frac{1}{4} y+\frac{3}{4} z=0} \\ {\frac{1}{4} x-\frac{1}{10} y+\frac{2}{5} z=-2} \\ {\frac{1}{8} x+\frac{1}{5} y-\frac{1}{8} z=2}\end{array}
$$

MB
Matt B.
Numerade Educator

Problem 33

For the following exercises, solve each system by Gaussian elimination.
$$
\begin{aligned} & \frac{4}{5} x-\frac{7}{8} y+\frac{1}{2} z=1 \\-\frac{4}{5} x-\frac{3}{4} y+\frac{1}{3} z &=-8 \\-\frac{2}{5} x-\frac{7}{8} y+\frac{1}{2} z &=-5 \end{aligned}
$$

Linh V.
Numerade Educator

Problem 34

For the following exercises, solve each system by Gaussian elimination.
$$
\begin{aligned}-\frac{1}{3} x-\frac{1}{8} y+\frac{1}{6} z &=-\frac{4}{3} \\-\frac{2}{3} x-\frac{7}{8} y+\frac{1}{3} z &=-\frac{23}{3} \\-\frac{1}{3} x-\frac{5}{8} y+\frac{5}{6} z &=0 \end{aligned}
$$

MB
Matt B.
Numerade Educator

Problem 35

For the following exercises, solve each system by Gaussian elimination.
$$
\begin{array}{l}{-\frac{1}{4} x-\frac{5}{4} y+\frac{5}{2} z=-5} \\ {-\frac{1}{2} x-\frac{5}{3} y+\frac{5}{4} z=\frac{55}{12}} \\ {-\frac{1}{3} x-\frac{1}{3} y+\frac{1}{3} z=\frac{5}{3}}\end{array}
$$

Linh V.
Numerade Educator

Problem 36

For the following exercises, solve each system by Gaussian elimination.
$$
\begin{array}{l}{\frac{1}{40} x+\frac{1}{60} y+\frac{1}{80} z=\frac{1}{100}} \\ {-\frac{1}{2} x-\frac{1}{3} y-\frac{1}{4} z=-\frac{1}{5}} \\ {\frac{3}{8} x+\frac{3}{12} y+\frac{3}{16} z=\frac{3}{20}}\end{array}
$$

MB
Matt B.
Numerade Educator

Problem 37

For the following exercises, solve each system by Gaussian elimination.
$$
\begin{array}{l}{0.1 x-0.2 y+0.3 z=2} \\ {0.5 x-0.1 y+0.4 z=8} \\ {0.7 x-0.2 y+0.3 z=8}\end{array}
$$

Linh V.
Numerade Educator

Problem 38

For the following exercises, solve each system by Gaussian elimination.
$$
\begin{array}{l}{0.2 x+0.1 y-0.3 z=0.2} \\ {0.8 x+0.4 y-1.2 z=0.1} \\ {1.6 x+0.8 y-2.4 z=0.2}\end{array}
$$

MB
Matt B.
Numerade Educator

Problem 39

For the following exercises, solve each system by Gaussian elimination.
$$
\begin{array}{l}{1.1 x+0.7 y-3.1 z=-1.79} \\ {2.1 x+0.5 y-1.6 z=-0.13} \\ {0.5 x+0.4 y-0.5 z=-0.07}\end{array}
$$

Linh V.
Numerade Educator

Problem 40

For the following exercises, solve each system by Gaussian elimination.
$$
\begin{array}{l}{0.5 x-0.5 y+0.5 z=10} \\ {0.2 x-0.2 y+0.2 z=4} \\ {0.1 x-0.1 y+0.1 z=2}\end{array}
$$

MB
Matt B.
Numerade Educator

Problem 41

For the following exercises, solve each system by Gaussian elimination.
$$
\begin{array}{l}{0.1 x+0.2 y+0.3 z=0.37} \\ {0.1 x-0.2 y-0.3 z=-0.27} \\ {0.5 x-0.1 y-0.3 z=-0.03}\end{array}
$$

Linh V.
Numerade Educator

Problem 42

For the following exercises, solve each system by Gaussian elimination.
$$
\begin{array}{l}{0.5 x-0.5 y-0.3 z=0.13} \\ {0.4 x-0.1 y-0.3 z=0.11} \\ {0.2 x-0.8 y-0.9 z=-0.32}\end{array}
$$

MB
Matt B.
Numerade Educator

Problem 43

For the following exercises, solve each system by Gaussian elimination.
$$
\begin{array}{l}{0.5 x+0.2 y-0.3 z=1} \\ {0.4 x-0.6 y+0.7 z=0.8} \\ {0.3 x-0.1 y-0.9 z=0.6}\end{array}
$$

Linh V.
Numerade Educator

Problem 44

For the following exercises, solve each system by Gaussian elimination.
$$
\begin{array}{l}{0.3 x+0.3 y+0.5 z=0.6} \\ {0.4 x+0.4 y+0.4 z=1.8} \\ {0.4 x+0.2 y+0.1 z=1.6}\end{array}
$$

MB
Matt B.
Numerade Educator

Problem 45

For the following exercises, solve each system by Gaussian elimination.
$$
\begin{array}{l}{0.8 x+0.8 y+0.8 z=2.4} \\ {0.3 x-0.5 y+0.2 z=0} \\ {0.1 x+0.2 y+0.3 z=0.6}\end{array}
$$

Linh V.
Numerade Educator

Problem 46

For the following exercises, solve the system for $x, y,$ and $z$
$$
\begin{array}{r}{x+y+z=3} \\ {\frac{x-1}{2}+\frac{y-3}{2}+\frac{z+1}{2}=0} \\ {\frac{x-2}{3}+\frac{y+4}{3}+\frac{z-3}{3}=\frac{2}{3}}\end{array}
$$

MB
Matt B.
Numerade Educator

Problem 47

For the following exercises, solve the system for $x, y,$ and $z$
$$
\begin{array}{l}{5 x-3 y-\frac{z+1}{2}=\frac{1}{2}} \\ {6 x+\frac{y-9}{2}+2 z=-3} \\ {\frac{x+8}{2}-4 y+z=4}\end{array}
$$

Linh V.
Numerade Educator

Problem 48

For the following exercises, solve the system for $x, y,$ and $z$
$$
\begin{array}{l}{\frac{x+4}{7}-\frac{y-1}{6}+\frac{z+2}{3}=1} \\ {\frac{x-2}{4}+\frac{y+1}{8}-\frac{z+8}{12}=0} \\ {\frac{x+6}{3}-\frac{y+2}{3}+\frac{z+4}{2}=3}\end{array}
$$

MB
Matt B.
Numerade Educator

Problem 49

For the following exercises, solve the system for $x, y,$ and $z$
$$
\begin{array}{l}{\frac{x-3}{6}+\frac{y+2}{2}-\frac{z-3}{3}=2} \\ {\frac{x+2}{4}+\frac{y-5}{2}+\frac{z+4}{2}=1} \\ {\frac{x+6}{2}-\frac{y-3}{2}+z+1=9}\end{array}
$$

Check back soon!

Problem 50

For the following exercises, solve the system for $x, y,$ and $z$
$$
\begin{aligned} \frac{x-1}{3}+\frac{y+3}{4}+\frac{z+2}{6} &=1 \\ 4 x+3 y-2 z &=11 \\ 0.02 x+0.015 y-0.01 z &=0.065 \end{aligned}
$$

MB
Matt B.
Numerade Educator

Problem 51

Three even numbers sum up to $108 .$ The smaller is half the larger and the middle number is $\frac{3}{4}$ the larger. What are the three numbers?

Linh V.
Numerade Educator

Problem 52

Three numbers sum up to 147 . The smallest number is half the middle number, which is half the largest
number. What are the three numbers?

MB
Matt B.
Numerade Educator

Problem 53

At a family reunion, there were only blood relatives, consisting of children, parents, and grandparents, in
attendance. There were 400 people total. There were twice as many parents as grandparents, and 50 more children than parents. How many children, parents, and grandparents were in attendance?

Linh V.
Numerade Educator

Problem 54

An animal shelter has a total of 350 animals comprised of cats, dogs, and rabbits. If the number of rabbits is 5 less than one-half the number of cats, and there are 20 more cats than dogs, how many of each animal are at the shelter?

MB
Matt B.
Numerade Educator

Problem 55

Your roommate, Sarah, offered to buy groceries for you and your other roommate. The total bill was
$\$ 82 .$ She forgot to save the individual receipts but remembered that your groceries were $\$ 0.05$ cheaper than half of her groceries, and that your other roommates groceries were $\$ 2.10$ more than your groceries. How much was each of your share of the groceries?

Linh V.
Numerade Educator

Problem 56

Your roommate, John, offered to buy household supplies for you and your other roommate. You live
near the border of three states, each of which has a different sales tax. The total amount of money spent
was $\$ 100.75$ . Your supplies were bought with 5$\%$ tax, John's with 8$\%$ tax, and your third roommates with 9$\%$ sales tax. The total amount of money spent without taxes is $\$ 93.50 .$ If your supplies before tax were $\$ 1$ more than half of what your third roommate's supplies were before tax, how much did each of you spend? Give your answer both with and without taxes.

MB
Matt B.
Numerade Educator

Problem 57

Three coworkers work for the same employer. Their jobs are warehouse manager, office manager, and
truck driver. The sum of the annual salaries of the warehouse manager and office manager is $\$ 82,000$ . The office manager makes $\$ 4,000$ more than the truck driver annually. The annual salaries of the warehouse manager and the truck driver total $\$ 78,000$ . What is the annual salary of each of the co-workers?

Linh V.
Numerade Educator

Problem 58

At a carnival, $\$ 2,914.25$ in receipts were taken at the end of the day. The cost of a child's ticket was $\$ 20.50$ , an adult ticket was $\$ 29.75$ , and a senior citizen ticket was $\$ 15.25$ . There were twice as many senior citizens as adults in attendance, and 20 more children than senior citizens. How many children, adult, and senior citizen tickets were sold?

MB
Matt B.
Numerade Educator

Problem 59

A local band sells out for their concert. They sell all $1,175$ tickets for a total purse of $\$ 28,112.50$ . The tickets were priced at $\$ 20$ for student tickets, $\$ 22.50$ for children, and $\$ 29$ for adult tickets. If the band sold twice as many adult as children tickets, how many of each type was sold?

Linh V.
Numerade Educator

Problem 60

In a bag, a child has 325 coins worth $\$ 19.50$ . There were three types of coins: pennies, nickels, and
dimes. If the bag contained the same number of nickels as dimes, how many of each type of coin was
in the bag?

MB
Matt B.
Numerade Educator

Problem 61

Last year, at Haven's Pond Car Dealership, for a particular model of BMW, Jeep, and Toyota, one
could purchase all three cars for a total of $\$ 140,000$ . This year, due to inflation, the same cars would cost $\$ 151,830$ . The cost of the BMW increased by $8 \%,$ the Jeep by 5$\%$ and the Toyota by 12$\% .$ If the price of last year's Jeep was $\$ 7,000$ less than the price of last year's BMW, what was the price of each of the three cars last year?

Linh V.
Numerade Educator

Problem 62

A recent college graduate took advantage of his business education and invested in three investments immediately after graduating. He invested $\$ 80,500$ into three accounts, one that paid 4$\%$ simple interest, one that paid 4$\%$ simple interest, one that paid 3$\frac{1}{8} \%$ simple interest, and one that paid 2$\frac{1}{2} \%$ simple interest. He earned $\$ 2,670$ interest at the end of one year. If the amount of the money invested in the second account was four times the amount invested in the third account, how much was invested in each account?

MB
Matt B.
Numerade Educator

Problem 63

You inherit one million dollars. You invest it all in three accounts for one year. The first account pays
3$\%$ compounded annually, the second account pays 4$\%$ compounded annually, and the third account pays 2$\%$ compounded annually. After one year, you earn $\$ 34,000$ in interest. If you invest four times the money into the account that pays 3$\%$ compared to 2$\%$ , how much did you invest in each account?

Linh V.
Numerade Educator

Problem 64

You inherit one hundred thousand dollars. You invest it all in three accounts for one year. The first account pays 4$\%$ compounded annually, the second account pays 3$\%$ compounded annually, and the third account pays 2$\%$ compounded annually. After one year, you earn $\$ 3,650$ in interest. If you invest five times the money in the account that pays 4 $\%$ compared to $3 \%,$ how much did you invest in each account?

MB
Matt B.
Numerade Educator

Problem 65

The top three countries in oil consumption in a certain year are as follows: the United States, Japan, and China. In millions of barrels per day, the three top countries consumed 39.8$\%$ of the worlds consumed oil. The United States consumed 0.7$\%$ more than four times China's consumption. The United States consumed 5$\%$ more than triple Japan's consumption. What percent of the world oil consumption did the United States, Japan, and China consume?"

Linh V.
Numerade Educator

Problem 66

The top three countries in oil production in the same year are Saudi Arabia, the United States, and Russia. In millions of barrels per day, the top three countries produced 31.4$\%$ of the world's produced oil. Saudi Arabia and the United States combined for 22.1$\%$ of the world's production, and Saudi Arabia produced 2$\%$ more oil than Russia. What percent of the world oil production did Saudi Arabia, the United States, and Russia produce?"

MB
Matt B.
Numerade Educator

Problem 67

The top three sources of oil imports for the United States in the same year were Saudi Arabia, Mexice,
and Canada. The three top countries accounted for 47$\%$ of oil imports. The United States imported 1.8$\%$ more from Saudi Arabia than they did from Mexico, and 1.7$\%$ more from Saudi Arabia than they did from Canada. What percent of the United States oil imports were from these three countries?""

Linh V.
Numerade Educator

Problem 68

The top three oil producers in the United States in a certain year are the Gulf of Mexico, Texas, and Alaska. The three regions were responsible for 64$\%$ of the United States oil production. The Gulf of Mexico and Texas combined for 47$\%$ of oil production. Texas produced 3$\%$ more than Alaska.
What percent of United States oil production came from these regions?""

MB
Matt B.
Numerade Educator

Problem 69

At one time, in the United States, 398 species of animals were on the endangered species list. The
top groups were mammals, birds, and fish, which comprised 55 $\%$ of the endangered species. Birds accounted for 0.7$\%$ more than fish, and fish accounted for 1.5$\%$ more than mammals. What
percent of the endangered species came from mammals, birds, and fish?

Linh V.
Numerade Educator

Problem 70

Meat consumption in the United States can be broken into three categories: red meat, poultry, and
fish. If fish makes up 4$\%$ less than one-quarter of poultry consumption, and red meat consumption is
18.2$\%$ higher than poultry consumption, what are the percentages of meat consumption?""l

MB
Matt B.
Numerade Educator