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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

stock therom F=(x2+y2) i-2xy J jaha x=a y=o y=b
Saurabh Reply
root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
Melanie Reply
simplify each radical by removing as many factors as possible (a) √75
Jason Reply
how is infinity bidder from undefined?
Karl Reply
what is the value of x in 4x-2+3
Vishal Reply
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
David Reply
v=lbh calculate the volume if i.l=5cm, b=2cm ,h=3cm
Haidar Reply
Need help with math
Peya
can you help me on this topic of Geometry if l help you
litshani
( cosec Q _ cot Q ) whole spuare = 1_cosQ / 1+cosQ
Aarav Reply
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So, the length of the guy wire can be found by evaluating √(90000+160000). What is the length of the guy wire?
Maxwell Reply
the indicated sum of a sequence is known as
Arku Reply
how do I attempted a trig number as a starter
Tumwe Reply
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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