Chapter 4, Systems of Linear Equations Video Solutions, Algebra for College Students

Section 3

Systems of Linear Equations in Three Variables

00:26

Problem 1

After reading this section, write out the answers to these questions. Use complete sentences.

What is a linear equation in three variables?

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00:32

Problem 2

After reading this section, write out the answers to these questions. Use complete sentences.

What is an ordered triple?

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00:45

Problem 3

After reading this section, write out the answers to these questions. Use complete sentences.

What is a solution to a system of linear equations in three variables?

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00:25

Problem 4

After reading this section, write out the answers to these questions. Use complete sentences.

How do we solve systems of linear equations in three variables?

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01:34

Problem 5

After reading this section, write out the answers to these questions. Use complete sentences.

What does the graph of a linear equation in three variables look like?

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00:39

Problem 6

After reading this section, write out the answers to these questions. Use complete sentences.

How are the planes positioned when a system of linear equations in three variables is independent?

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01:04

Problem 7

Solve each system of equations.
$$\begin{array}{r}x+y+z=9 \\y+z=7 \\z=4\end{array}$$

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01:14

Problem 8

Solve each system of equations.
$$\begin{aligned}x+y-z &=4 \\y &=6 \\y+z &=13\end{aligned}$$

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01:45

Problem 9

Solve each system of equations.
$$\begin{aligned}&x+y+z=10\\&\begin{array}{ll}x-y & =-1 \\x+y & =5\end{array}\end{aligned}$$

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01:40

Problem 10

Solve each system of equations.
$$\begin{aligned}x+y-z &=6 \\y+z &=11 \\y-z &=3\end{aligned}$$

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02:53

Problem 11

Solve each system of equations.
$$\begin{aligned}&x+y+z=6\\&x-y+z=2\\&x-y-z=-4\end{aligned}$$

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02:06

Problem 12

Solve each system of equations.
$$\begin{aligned}&x+y+z=0\\&x+y-z=2\\&x-y+z=0\end{aligned}$$

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02:52

Problem 13

Solve each system of equations.

$$\begin{array}{l}x+y+z=2 \\x+2 y-z=6 \\2 x+y-z=5\end{array}$$

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04:50

Problem 14

Solve each system of equations.
$$\begin{aligned}2 x-y+3 z &=14 \\x+y-2 z &=-5 \\3 x+y-z &=2\end{aligned}$$

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03:57

Problem 15

Solve each system of equations.

$$\begin{aligned}&x-2 y+4 z=3\\&x+3 y-2 z=6\\&x-4 y+3 z=-5\end{aligned}$$

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02:11

Problem 16

Solve each system of equations.

$$\begin{aligned}2 x+3 y+z &=13 \\-3 x+2 y+z &=-4 \\4 x-4 y+z &=5\end{aligned}$$

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03:31

Problem 17

Solve each system of equations.

$$\begin{aligned}2 x-y+z &=10 \\3 x-2 y-2 z &=7 \\x-3 y-2 z &=10\end{aligned}$$

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03:42

Problem 18

Solve each system of equations.

$$\begin{aligned}x-3 y+2 z &=-11 \\2 x-4 y+3 z &=-15 \\3 x-5 y-4 z &=5\end{aligned}$$

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02:00

Problem 19

Solve each system of equations.

$$\begin{aligned}2 x-3 y+z &=-9 \\-2 x+y-3 z &=7 \\x-y+2 z &=-5\end{aligned}$$

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04:09

Problem 20

Solve each system of equations.

$$\begin{aligned}3 x-4 y+z &=19 \\2 x+4 y+z &=0 \\x-2 y+5 z &=17\end{aligned}$$

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03:11

Problem 21

Solve each system of equations.

$$\begin{aligned}&2 x-5 y+2 z=16\\&3 x+2 y-3 z=-19\\&4 x-3 y+4 z=18\end{aligned}$$

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03:05

Problem 22

Solve each system of equations.

$$\begin{array}{r}-2 x+3 y-4 z=3 \\3 x-5 y+2 z=4 \\-4 x+2 y-3 z=0\end{array}$$

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01:41

Problem 23

Solve each system of equations.

$$\begin{aligned}x+y &=4 \\y-z &=-2 \\x+y+z &=9\end{aligned}$$

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01:29

Problem 24

Solve each system of equations.

$$\begin{aligned}x+y-z &=0 \\x-y &=-2 \\y+z &=10\end{aligned}$$

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01:37

Problem 25

Solve each system of equations.

$$\begin{aligned}x+y &=7 \\y-z &=-1 \\x &+3 z=18\end{aligned}$$

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01:43

Problem 26

Solve each system of equations.

$$\begin{aligned}2 x-y &=-8 \\y+3 z &=22 \\x-z &=-8\end{aligned}$$

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01:27

Problem 27

Solve each system.

$$\begin{aligned}x+y-z &=2 \\-x-y+z &=-2 \\2 x+2 y-2 z &=4\end{aligned}$$

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00:54

Problem 28

Solve each system.

$$\begin{array}{r}x+y+z=1 \\2 x+2 y+2 z=2 \\4 x+4 y+4 z=4\end{array}$$

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00:49

Problem 29

Solve each system.
$$\begin{aligned}&x+y-z=2\\&x+y+z=8\\&x+y-z=6\end{aligned}$$

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00:48

Problem 30

Solve each system.
$$\begin{aligned}x+y+z &=6 \\2 z+2 y+2 z &=9 \\3 x+3 y+3 z &=12\end{aligned}$$

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00:41

Problem 31

Solve each system.
$$\begin{aligned}x+y+z &=9 \\x+y &=5 \\z &=1\end{aligned}$$

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01:22

Problem 32

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

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01:01

Problem 33

Solve each system.
$$\begin{array}{r}x-y+2 z=3 \\2 x+y-z=5 \\3 x-3 y+6 z=4\end{array}$$

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01:00

Problem 34

Solve each system.
$$\begin{array}{r}4 x-2 y-2 z=5 \\2 x-y-z=7 \\-4 x+2 y+2 z=6\end{array}$$

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01:43

Problem 35

Solve each system.

$$\begin{array}{l}2 x-4 y+6 z=12 \\6 x-12 y+18 z=36 \\-x+2 y-3 z=-6\end{array}$$

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01:03

Problem 36

Solve each system.
$$\begin{aligned}3 x-y+z &=5 \\9 x-3 y+3 z &=15 \\-12 x+4 y-4 z &=-20\end{aligned}$$

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01:06

Problem 37

Solve each system.
$$\begin{aligned}x-y &=3 \\y+z &=8 \\2 x &=7\end{aligned}$$

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01:46

Problem 38

Solve each system.
$$\begin{aligned}2 x-y &=6 \\2 y+z &=-4 \\8 x &=3\end{aligned}$$

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01:11

Problem 39

Solve each system.

$$\begin{aligned}0.10 x+0.08 y-0.04 z &=3 \\5 x+4 y-2 z &=150 \\0.3 x+0.24 y-0.12 z &=9\end{aligned}$$

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00:52

Problem 40

Solve each system.

$$\begin{aligned}0.06 x-0.04 y+z &=6 \\3 x-2 y+50 z &=300 \\0.03 x-0.02 y+0.5 z &=3\end{aligned}$$

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Problem 41

Use a calculator to solve each system.

$$\begin{aligned}3 x+2 y-0.4 z &=0.1 \\3.7 x-0.2 y+0.05 z &=0.41 \\-2 x+3.8 y-2.1 z &=-3.26\end{aligned}$$

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Problem 42

Use a calculator to solve each system.

$$\begin{aligned}3 x-0.4 y+9 z &=1.668 \\0.3 x+5 y-8 z &=-0.972 \\5 x-4 y-8 z &=1.8\end{aligned}$$

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01:23

Problem 43

Solve each problem by using a system of three equations in
three unknowns.

Three cars. The town of Springfield purchased a Chevrolet, a Ford, and a Toyota for a total of 66,000 dollars. The Ford was 2,000 dollars more than the Chevrolet and the Toyota was 2,000 dollars more than the Ford. What was the price of each car?

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01:21

Problem 44

Solve each problem by using a system of three equations in
three unknowns.

Buying texts. Melissa purchased an English text, a math text, and a chemistry text for a total of 276 .$ The English text was 20 dollars more than the math text and the chemistry text was twice the price of the math text. What was the price of each text?

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03:25

Problem 45

Solve each problem by using a system of three equations in
three unknowns.

Three-day drive. In three days, Carter drove 2196 miles in 36 hours behind the wheel. The first day he averaged 64 \mathrm{mph}, the second day 62 \mathrm{mph}, and the third day 58 \mathrm{mph}

If he drove 4 more hours on the third day than on the first day, then how many hours did he drive each day?

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04:26

Problem 46

Solve each problem by using a system of three equations in
three unknowns.

In three days, Katy traveled 146 miles down the Mississippi River in her kayak with 30 hours of paddling. The first day she averaged 6{mph}, the second day 5{mph}, and the third day 4 {mph}. If her distance on the third day was equal to her distance on the first day, then for how many hours did she paddle each day?

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02:44

Problem 47

Solve each problem by using a system of three equations in
three unknowns.

Ann invested a total of 12,000 dollars in stocks,
bonds, and a mutual fund. She received a 10 \% return on her stock investment, an 8 \% return on her bond investment, and a 12 \% return on her mutual fund. Her total return was 1230 dollars. If the total investment in stocks and bonds equaled her mutual fund investment, then how much did she invest in each?

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02:20

Problem 48

Solve each problem by using a system of three equations in
three unknowns.

Paranoia. Fearful of a bank failure, Norman split his life savings of 60,000 dollars among three banks. He received 5 \%,6 \%, and 7 \% on the three deposits. In the account earning 7 \% interest, he deposited twice as much as in the account earning 5 \% interest. If his total earnings were 3760 dollars, then how much did he deposit in each account?

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01:46

Problem 49

Solve each problem by using a system of three equations in
three unknowns.
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Anna, Bob, and Chris will not disclose their weights but agree to be weighed in pairs. Anna and Bob together weigh 226 pounds. Bob and Chris together weigh 210 pounds. Anna and Chris together weigh 200 pounds. How much does each student weigh?

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01:55

Problem 50

Solve each problem by using a system of three equations in
three unknowns.

On Monday Headley paid 1.70 dollars for two cups
of coffee and one doughnut, including the tip. On Tuesday he paid 1.65 dollars for two doughnuts and a cup of coffee, including the tip. On Wednesday he paid 1.30 dollars for one coffee and one doughnut, including the tip. If he always tips the same amount, then what is the amount of each item?

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02:23

Problem 51

Solve each problem by using a system of three equations in
three unknowns.

Three coins. Nelson paid 1.75 dollars for his lunch with
13 coins, consisting of nickels, dimes, and quarters. If the number of dimes was twice the number of nickels, then how many of each type of coin did he use?

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03:20

Problem 52

Solve each problem by using a system of three equations in
three unknowns.

Harry has 2.25 dollars in nickels, dimes, and quarters. If he had twice as many nickels, half as many dimes, and the same number of quarters, he would have 2.50 dollars . If he has 27 coins altogether, then how many of each does he have?

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02:55

Problem 53

Solve each problem by using a system of three equations in
three unknowns.

Working overtime. To make ends meet, Ms. Farns by works three jobs. Her total income last year was 48,000 dollars. Her income from teaching was just 6000 dollars more than her income from house painting. Royalties from her textbook sales were one-seventh of the total money she received from teaching and house painting. How much did she make from each source last year?

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03:16

Problem 54

Solve each problem by using a system of three equations in
three unknowns.

Salvador's Fruit Mart sells variety packs. The small pack contains three bananas, two apples, and one orange for 1.80 dollars . The medium pack contains four bananas, three apples, and three oranges for 3.05 dollars. The family size contains six bananas, five apples, and four oranges for 4.65 dollars . What price should Salvador charge for his lunch-box special that consists of one banana, one apple, and one orange?

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03:45

Problem 55

Solve each problem by using a system of three equations in
three unknowns.

Three generations. Edwin, his father, and his grandfather have an average age of 53 dollars. One-half of his grandfather's age, plus one-third of his father's age, plus one-fourth of Edwin's age is 65 dollars . If 4 years ago, Edwin's grandfather was four times as old as Edwin, then how old are they all now?

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03:30

Problem 56

Solve each problem by using a system of three equations in
three unknowns.

The sum of the digits of a three-digit number is 11 dollars . If the digits are reversed, the new number is 46 more than five times the old number. If the hundreds digit plus twice the tens digit is equal to the units digit, then what is the number?

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Problem 57

Exploration
Draw diagrams showing the possible ways to position three planes in three-dimensional space.

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01:03

Problem 58

Discussion
Make up a system of three linear equations in three variables for which the solution set is $\{(0,0,0)\} .$ A system with this solution set is called a homogeneous system. Why do you think it is given that name?

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