# College Algebra (Open Stax)

## Educators

MB

### Problem 1

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

Linh V.

### 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.

### 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.

### Problem 4

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

MB
Matt B.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### Problem 55

Your roommate, Sarah, offered to buy groceries for you and your other roommate. The total bill was

MB
Matt B.

### 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.

### Problem 61

Last year, at Haven's Pond Car Dealership, for a particular model of BMW, Jeep, and Toyota, one

Linh V.