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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

the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
test for convergence the series 1+x/2+2!/9x3
success Reply
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
Lhorren Reply
100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
jancy Reply
answer
Ajith
exponential series
Naveen
what is subgroup
Purshotam Reply
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
Macmillan Reply
e power cos hyperbolic (x+iy)
Vinay Reply
10y
Michael
tan hyperbolic inverse (x+iy)=alpha +i bita
Payal Reply
prove that cos(π/6-a)*cos(π/3+b)-sin(π/6-a)*sin(π/3+b)=sin(a-b)
Tejas Reply
why {2kπ} union {kπ}={kπ}?
Huy Reply
why is {2kπ} union {kπ}={kπ}? when k belong to integer
Huy
if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41
Trilochan Reply
what is complex numbers
Ayushi Reply
Please you teach
Dua
Yes
ahmed
Thank you
Dua
give me treganamentry question
Anshuman Reply
Solve 2cos x + 3sin x = 0.5
shobana 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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