A system of linear equations allows the relationship between two or more linear equations to be compared and analyzed. 4.1 - Systems of Linear Equations

3 3 0

2

x y

x y

A system of linear equations allows the relationship between two or more linear equations to be compared and analyzed.

0

2 10

x y

x y

7 1

4

y x

y

72

92

432

y x

y x

y x

4.1 - Systems of Linear Equations in Two Variables

Determine whether (3, 9) is a solution of the following system.

5 2 3

3

x y

y x

5 3 2 39

15 18 3 3 3

9 33

9 9

Both statements are true, therefore (3, 9) is a solution to the given system of linear equations.


Determine whether (3, -2) is a solution of the following system.

2 8

3 4

x y

x y

2 3 2 8

6 2 8 8 8

3 23 4

3 6 4

Both statements are not true, therefore (3, -2) is not a solution to the given system of linear equations.

3 4


Solving Systems of Linear Equations by Graphing

4 1

3

y x

y

: 1,3Solution

3 3 3 4 1 1

3 3


Solving Systems of Linear Equations by Graphing

2 0

3x y

x y

: 1, 2Solution

3 3

2 1 2 0

0 0

3y x

1 2 3

2y x


Solving Systems of Linear Equations by the Addition Method

9357 3328

126315

2036610 2950627

9141217


(Also referred to as the Elimination Method)

64

13

yx

yx

13 yx

113 y

64 yx

13 yx

13 y2 y

2y

2,1Solution

77 x

77 x

1x




3210

145

yx

yx

145 yx

145

15

y

32102 yx

145 yx

141 y

24 y

2

1,

5

1Solution

525 x5

1x

145 yx

6420 yx 2

1y




943

652

yx

yx

6523 yx

6052 x

9432 yx

652 yx

62 x3x

0,3

Solution

07 y

0y

18156 yx

1886 yx

4.1 - Systems of Linear Equations in Two VariablesSolving Systems of Linear Equations by the Addition Method


20148

1074

yx

yx

10742 yx

20148 yx

00

20148 yx20148 yx

True Statement


Solution: All reals

Lines are the same




23

593

yx

yx

233 yx593 yx

10

593 yx693 yx

lines are parallel

False Statement

No Solution



Solving Systems of Linear Equations by Substitution

5 2 3

3

x y

y x

325 yx

3325 xx

3x

365 xx

3 x

xy 3

33y

9y

9,3Solution


Solving Systems of Linear Equations by Substitution

2 8

3 4

x y

x y

43 yx

43 yx0y

82 yx

07 y

82 yx

802 x

4x

0,4Solution

8432 yy

886 yy

887 y82 x


Example4.1 - Systems of Linear Equations in Two Variables

LCD: 6

LCD: 15

Solution(2 ,−1)

A first number is seven greater than a second number. Twice the first number is four more than three times the second number. What are the numbers?

4.3 - Applications

4372 yy

7yx

Substitution Method

7yx

1017,4y10

432 yx

43142 yy 710x

1st number is x, 2nd number is y

4y2 y2 17xSolution

Two trains leave Tulsa, one traveling north and the other south. After four hours, they are 376 miles apart. If one train is traveling ten miles per hour faster than the other, what is the speed of each train?

37644 yx10yx

Substitution Method

1042x

376408 y

10yx

3764404 yy

3764104 yy

52x

42y3368 ymph

mph

Train Rate Time Distance

North

Southx 4y

4x4y4

4.3 - Applications

A boat can travel 20 miles down-stream in 2 hours. It can travel 18 upstream in 3 hours. What is the speed of the boat in still water and the speed of the current?

Elimination Method

Rate Time Distance

w/current

Against curr.𝑥+𝑦 2𝑥− 𝑦

20183

4.3 - Applications

(𝑥+𝑦 ) 2=20(𝑥− 𝑦 ) 3=18

𝑥+𝑦=10𝑥− 𝑦=62 𝑥=16𝑥=8

𝑥+𝑦=108+𝑦=10𝑦=2

Boat speed: 8 mphCurrent speed: 2 mph

Boat speed: x Current speed: y

One solution contains 20% acid and a second solution contains 50% acid. How many ounces of each solution should be mixed in order to have sixty ounces of a 30% solution?

3.0605.02.0 yx

60 yx

185.02.0 yx

Solution Ounces Decimal Pure Acid

20%

50%

30%

x 0.2y

0.2x0.5y0.5

60 0.3 (60)(0.3)

4.3 - Applications

18052 yx

One solution contains 20% acid and a second solutions contains 50% acid. How many ounces of each solution should be mixed in order to have sixty ounces of a 30% solution?

18052 yx

18052 yx

Elimination Method

6020 x

18052 yx

60 yx

602 yx

12022 yx

40x

603 y

60 yx

solutiontheofounces %2040

603 y20y

solutiontheofounces %5020

4.3 - Applications

For a particular show the price of an adult ticket is $2.00 and a child's ticket is $1.50. A total of 300 tickets were sold for $525. How many adult and children’s tickets were sold?

300CA

7550 C.

525512 CA .

Tickets Type Price Cost

Adult

Child

Total

A $2.00C

2A1.5C$1.50

300 $525

4.3 - Applications

60022 CA525512 CA .

150C300150A150A

150 Adult tickets150 Children’s tickets

Elimination Method

7550 C.

The value of 12 coins is $1.20. The coin are nickels, dimes and quarters. The number of dimes is two more than twice the number of nickels. How many nickels, dimes and quarters are there?

Elimination Method12 qdn12025105 qdn

4.3 - Applications

22 nd

𝑛+2𝑛+2+𝑞=125𝑛+10 (2𝑛+2)+25𝑞=120

3𝑛+2+𝑞=125𝑛+20𝑛+20+25𝑞=120

3𝑛+𝑞=1025𝑛+25𝑞=100

−75𝑛−25𝑞=−25025𝑛+25𝑞=100−50𝑛=−150

𝑛=3𝑑=2 (3 )+2

𝑑=83+8+𝑞=12

𝑞=1

nickels

dimes

quarter

4.4 – Systems of Linear InequalitiesGraphing Inequalities in Two Variables

Graph the solution.

2

13

2

xy

xy

Graphing Inequalities in Two Variables

Graph the solution.

2

13

2

xy

xy

4.4 – Systems of Linear Inequalities


Graph the solution.

2

22

1

x

xy



Graph the solution.

2

22

1

x

xy



Graph the solution.

93

032

yx

xy

32xy

33

1 xy


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A system of linear equations allows the relationship between two or more linear equations to be compared and analyzed. 4.1 - Systems of Linear Equations