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Algebraic

For the following exercises, determine whether the ordered triple given is the solution to the system of equations.

2 x −6 y + 6 z = −12    x + 4 y + 5 z = −1 −x + 2 y + 3 z = −1 and ( 0 , 1 , −1 )

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  6 x y + 3 z = 6 3 x + 5 y + 2 z = 0             x + y = 0 and ( 3 , −3 , −5 )

No

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6 x −7 y + z = 2 −x y + 3 z = 4    2 x + y z = 1 and ( 4 , 2 , −6 )

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      x y = 0        x z = 5 x y + z = −1 and ( 4 , 4 , −1 )

Yes

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  −x y + 2 z = 3   5 x + 8 y −3 z = 4 −x + 3 y −5 z = −5 and ( 4 , 1 , −7 )

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For the following exercises, solve each system by substitution.

3 x −4 y + 2 z = −15    2 x + 4 y + z = 16   2 x + 3 y + 5 z = 20

( −1 , 4 , 2 )

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5 x −2 y + 3 z = 20 2 x −4 y −3 z = −9    x + 6 y −8 z = 21

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  5 x + 2 y + 4 z = 9 −3 x + 2 y + z = 10   4 x −3 y + 5 z = −3

( 85 107 , 312 107 , 191 107 )

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4 x −3 y + 5 z = 31 x + 2 y + 4 z = 20     x + 5 y −2 z = −29

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   5 x −2 y + 3 z = 4 −4 x + 6 y −7 z = −1      3 x + 2 y z = 4

( 1 , 1 2 , 0 )

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    4 x + 6 y + 9 z = 0 −5 x + 2 y −6 z = 3     7 x −4 y + 3 z = −3

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For the following exercises, solve each system by Gaussian elimination.

     2 x y + 3 z = 17 −5 x + 4 y −2 z = −46             2 y + 5 z = −7

( 4 , −6 , 1 )

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5 x −6 y + 3 z = 50       x + 4 y = 10           2 x z = 10

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     2 x + 3 y −6 z = 1 −4 x −6 y + 12 z = −2        x + 2 y + 5 z = 10

( x , 1 27 ( 65 −16 x ) , x + 28 27 )

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  4 x + 6 y −2 z = 8   6 x + 9 y −3 z = 12 −2 x −3 y + z = −4

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2 x + 3 y −4 z = 5 −3 x + 2 y + z = 11 x + 5 y + 3 z = 4

( 45 13 , 17 13 , −2 )

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10 x + 2 y −14 z = 8    −x −2 y −4 z = −1 −12 x −6 y + 6 z = −12

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     x + y + z = 14        2 y + 3 z = −14 −16 y −24 z = −112

No solutions exist

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   5 x −3 y + 4 z = −1 −4 x + 2 y −3 z = 0   −x + 5 y + 7 z = −11

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        x + y + z = 0     2 x y + 3 z = 0               x z = 0

( 0 , 0 , 0 )

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3 x + 2 y −5 z = 6 5 x −4 y + 3 z = −12 4 x + 5 y −2 z = 15

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     x + y + z = 0 2 x y + 3 z = 0            x z = 1

( 4 7 , 1 7 , 3 7 )

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3 x 1 2 y z = 1 2           4 x + z = 3      x + 3 2 y = 5 2

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     6 x −5 y + 6 z = 38 1 5 x 1 2 y + 3 5 z = 1   −4 x 3 2 y z = −74

( 7 , 20 , 16 )

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1 2 x 1 5 y + 2 5 z = 13 10 1 4 x 2 5 y 1 5 z = 7 20 1 2 x 3 4 y 1 2 z = 5 4

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1 3 x 1 2 y 1 4 z = 3 4 1 2 x 1 4 y 1 2 z = 2 1 4 x 3 4 y 1 2 z = 1 2

( −6 , 2 , 1 )

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1 2 x 1 4 y + 3 4 z = 0 1 4 x 1 10 y + 2 5 z = −2 1 8 x + 1 5 y 1 8 z = 2

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   4 5 x 7 8 y + 1 2 z = 1 4 5 x 3 4 y + 1 3 z = −8 2 5 x 7 8 y + 1 2 z = −5

( 5 , 12 , 15 )

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1 3 x 1 8 y + 1 6 z = 4 3 2 3 x 7 8 y + 1 3 z = 23 3 1 3 x 5 8 y + 5 6 z = 0

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1 4 x 5 4 y + 5 2 z = −5 1 2 x 5 3 y + 5 4 z = 55 12 1 3 x 1 3 y + 1 3 z = 5 3

( −5 , −5 , −5 )

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1 40 x + 1 60 y + 1 80 z = 1 100 1 2 x 1 3 y 1 4 z = 1 5 3 8 x + 3 12 y + 3 16 z = 3 20

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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

( 10 , 10 , 10 )

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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

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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

( 1 2 , 1 5 , 4 5 )

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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

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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

( 1 2 , 2 5 , 4 5 )

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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

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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

( 2 , 0 , 0 )

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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

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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

( 1 , 1 , 1 )

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Extensions

For the following exercises, solve the system for x , y , and z .

                  x + y + z = 3 x −1 2 + y −3 2 + z + 1 2 = 0 x −2 3 + y + 4 3 + z −3 3 = 2 3

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5 x −3 y z + 1 2 = 1 2 6 x + y −9 2 + 2 z = −3    x + 8 2 −4 y + z = 4

( 128 557 , 23 557 , 28 557 )

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x + 4 7 y −1 6 + z + 2 3 = 1 x −2 4 + y + 1 8 z + 8 12 = 0 x + 6 3 y + 2 3 + z + 4 2 = 3

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x −3 6 + y + 2 2 z −3 3 = 2 x + 2 4 + y −5 2 + z + 4 2 = 1 x + 6 2 y −3 2 + z + 1 = 9

( 6 , −1 , 0 )

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     x −1 3 + y + 3 4 + z + 2 6 = 1                  4 x + 3 y −2 z = 11 0.02 x + 0.015 y −0.01 z = 0.065

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Real-world applications

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

24, 36, 48

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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?

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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?

70 grandparents, 140 parents, 190 children

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Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
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what is inorganic
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Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
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Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
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Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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