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5 x y = 4 x + 6 y = 2 and ( 4 , 0 )

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−3 x 5 y = 13   x + 4 y = 10 and ( −6 , 1 )

Yes

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3 x + 7 y = 1 2 x + 4 y = 0 and ( 2 , 3 )

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−2 x + 5 y = 7    2 x + 9 y = 7 and ( −1 , 1 )

Yes

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x + 8 y = 43 3 x −2 y = −1 and ( 3 , 5 )

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For the following exercises, solve each system by substitution.

   x + 3 y = 5 2 x + 3 y = 4

( −1 , 2 )

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  3 x −2 y = 18 5 x + 10 y = −10

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4 x + 2 y = −10 3 x + 9 y = 0

( −3 , 1 )

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2 x + 4 y = −3.8 9 x −5 y = 1.3

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2 x + 3 y = 1.2 3 x 6 y = 1.8

( 3 5 , 0 )

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    x −0.2 y = 1 −10 x + 2 y = 5

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     3 x + 5 y = 9 30 x + 50 y = −90

No solutions exist.

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  −3 x + y = 2 12 x −4 y = −8

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1 2 x + 1 3 y = 16 1 6 x + 1 4 y = 9

( 72 5 , 132 5 )

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1 4 x + 3 2 y = 11 1 8 x + 1 3 y = 3

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For the following exercises, solve each system by addition.

−2 x + 5 y = −42    7 x + 2 y = 30

( 6 , −6 )

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6 x −5 y = −34 2 x + 6 y = 4

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     5 x y = −2.6 −4 x −6 y = 1.4

( 1 2 , 1 10 )

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7 x −2 y = 3 4 x + 5 y = 3.25

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  −x + 2 y = −1 5 x −10 y = 6

No solutions exist.

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       7 x + 6 y = 2 −28 x −24 y = −8

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5 6 x + 1 4 y = 0 1 8 x 1 2 y = 43 120

( 1 5 , 2 3 )

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   1 3 x + 1 9 y = 2 9 1 2 x + 4 5 y = 1 3

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−0.2 x + 0.4 y = 0.6             x −2 y = −3

( x , x + 3 2 )

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−0.1 x + 0.2 y = 0.6       5 x −10 y = 1

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For the following exercises, solve each system by any method.

5 x + 9 y = 16    x + 2 y = 4

( −4 , 4 )

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6 x −8 y = −0.6 3 x + 2 y = 0.9

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5 x −2 y = 2.25 7 x −4 y = 3

( 1 2 , 1 8 )

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   x 5 12 y = 55 12 −6 x + 5 2 y = 55 2

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7 x −4 y = 7 6 2 x + 4 y = 1 3

( 1 6 , 0 )

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3 x + 6 y = 11 2 x + 4 y = 9

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       7 3 x 1 6 y = 2 21 6 x + 3 12 y = −3

( x , 2 ( 7 x −6 ) )

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1 2 x + 1 3 y = 1 3 3 2 x + 1 4 y = 1 8

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2.2 x + 1.3 y = −0.1 4.2 x + 4.2 y = 2.1

( 5 6 , 4 3 )

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  0.1 x + 0.2 y = 2 0.35 x −0.3 y = 0

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Graphical

For the following exercises, graph the system of equations and state whether the system is consistent, inconsistent, or dependent and whether the system has one solution, no solution, or infinite solutions.

3 x y = 0.6 x −2 y = 1.3

Consistent with one solution

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x + 2 y = 4   2 x −4 y = 1

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   x + 2 y = 7 2 x + 6 y = 12

Consistent with one solution

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3 x −5 y = 7    x −2 y = 3

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   3 x −2 y = 5 −9 x + 6 y = −15

Dependent with infinitely many solutions

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Technology

For the following exercises, use the intersect function on a graphing device to solve each system. Round all answers to the nearest hundredth.

    0.1 x + 0.2 y = 0.3 −0.3 x + 0.5 y = 1

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−0.01 x + 0.12 y = 0.62 0.15 x + 0.20 y = 0.52

( −3.08 , 4.91 )

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0.5 x + 0.3 y = 4 0.25 x −0.9 y = 0.46

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0.15 x + 0.27 y = 0.39 −0.34 x + 0.56 y = 1.8

( −1.52 , 2.29 )

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−0.71 x + 0.92 y = 0.13 0.83 x + 0.05 y = 2.1

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Extensions

For the following exercises, solve each system in terms of A , B , C , D , E , and F where A F are nonzero numbers. Note that A B and A E B D .

x + y = A x y = B

( A + B 2 , A B 2 )

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A x + y = 0 B x + y = 1

( −1 A B , A A B )

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A x + B y = C D x + E y = F

( C E B F B D A E , A F C D B D A E )

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Real-world applications

For the following exercises, solve for the desired quantity.

A stuffed animal business has a total cost of production C = 12 x + 30 and a revenue function R = 20 x . Find the break-even point.

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A fast-food restaurant has a cost of production C ( x ) = 11 x + 120 and a revenue function R ( x ) = 5 x . When does the company start to turn a profit?

They never turn a profit.

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A cell phone factory has a cost of production C ( x ) = 150 x + 10 , 000 and a revenue function R ( x ) = 200 x . What is the break-even point?

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A musician charges C ( x ) = 64 x + 20,000 , where x is the total number of attendees at the concert. The venue charges $80 per ticket. After how many people buy tickets does the venue break even, and what is the value of the total tickets sold at that point?

( 1 , 250 , 100 , 000 )

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A guitar factory has a cost of production C ( x ) = 75 x + 50,000. If the company needs to break even after 150 units sold, at what price should they sell each guitar? Round up to the nearest dollar, and write the revenue function.

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Questions & Answers

Find the possible value of 8.5 using moivre's theorem
Reuben Reply
which of these functions is not uniformly cintinuous on (0, 1)? sinx
Pooja Reply
which of these functions is not uniformly continuous on 0,1
Basant Reply
solve this equation by completing the square 3x-4x-7=0
Jamiz Reply
X=7
Muustapha
=7
mantu
x=7
mantu
3x-4x-7=0 -x=7 x=-7
Kr
x=-7
mantu
9x-16x-49=0 -7x=49 -x=7 x=7
mantu
what's the formula
Modress
-x=7
Modress
new member
siame
what is trigonometry
Jean Reply
deals with circles, angles, and triangles. Usually in the form of Soh cah toa or sine, cosine, and tangent
Thomas
solve for me this equational y=2-x
Rubben Reply
what are you solving for
Alex
solve x
Rubben
you would move everything to the other side leaving x by itself. subtract 2 and divide -1.
Nikki
then I got x=-2
Rubben
it will b -y+2=x
Alex
goodness. I'm sorry. I will let Alex take the wheel.
Nikki
ouky thanks braa
Rubben
I think he drive me safe
Rubben
how to get 8 trigonometric function of tanA=0.5, given SinA=5/13? Can you help me?m
Pab Reply
More example of algebra and trigo
Stephen Reply
What is Indices
Yashim Reply
If one side only of a triangle is given is it possible to solve for the unkown two sides?
Felix Reply
cool
Rubben
kya
Khushnama
please I need help in maths
Dayo Reply
Okey tell me, what's your problem is?
Navin
the least possible degree ?
Dejen Reply
(1+cosA)(1-cosA)=sin^2A
BINCY Reply
good
Neha
why I'm sending you solved question
Mirza
Teach me abt the echelon method
Khamis
exact value of cos(π/3-π/4)
Ankit Reply
What is differentiation?
Intakhab Reply

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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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