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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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Practice Key Terms 1

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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