systems of linear equations block 44. system of linear equations a system of equations is a set or...

Systems of Linear Equations

Block 44

System of Linear Equations

A system of equations is a set or collection of equations that you deal with all together at once.

Linear equations (ones that graph as straight lines) are simpler than non-linear equations

The simplest linear system is one with two equations and two variables.

Graph of a Linear Equation

Graph of y = 3x – 2

x y

1 1

0 -2

-1 -5

Graph of a Linear Equation

Graph of y = –x – 6

x y

1 -7

0 -6

-1 -5


Graph of y = 3x – 2 & y = –x – 6

x y

1 1

0 -2

-1 -5

x y

1 -7

0 -6

-1 -5


Graph of y = 3x – 2 & y = –x – 6

x y

1 1

0 -2

-1 -5

x y

1 -7

0 -6

-1 -5

Solution is (-1, -5)

Practice Solving Systems of Linear Equations

Solve by Graphing the following systems of linear equations (see worksheet #1):

#1



#2



#3



#4



#5

Solving Systems of Linear Equations

Substitution Method:

2x – 3y = –24x + y = 24

Choose 2nd equation: 4x + y = 24 Rewrite with single variable: y = 24 – 4xSubstitute into 1st equation: 2x – 3(24 – 4x) = –2



2x – 3y = –24x + y = 24

Simplify: 2x – 72 + 12x = –2 14x – 72 = -2

14x = 70 x = 5



2x – 3y = –24x + y = 24

Substitute x = 5 into either equation:

4x + y = 24 4(5) + y = 24 20 + y = 24

y = 24 – 20

y = 4



2x – 3y = –24x + y = 24

The solution is the ordered pair (5, 4).


Solve by Substitution the following systems of linear equations (see worksheet #2):

#1



#2



#3



#4



#5

Solving an Equation

Addition or Elimination Method:

Example: x + 6 = 11

-6 -6

x = 5


Addition or Elimination Method-easy:

2x + y = 93x – y = 16

Add: 5x = 25

Simplify: x = 5

Substitute: 2(5) + y = 9

10 + y = 9

y = -1


Addition or Elimination Method - easy:

2x + y = 93x – y = 16

Solution is (5, -1)


Addition or Elimination Method – medium:

2x – y = 93x + 4y = –14

Multiply 1st by 4: 8x – 4y = 36

8x – 4 y = 363x + 4y = –14



8x – 4 y = 363x + 4y = –14


Add: 11x = 22

Simplify: x = 2

Substitute: 2(2) – y = 9 4 – y = 9

-y = 5 or y = -5



2x – y = 93x + 4y = –14

Solution is (2, -5)


Addition or Elimination Method – hard:

4x – 3y = 25 –3x + 8y = 10


Multiply 2nd by 4: -12x + 32y = 40

Solving Systems of Linear EquationsAddition or Elimination Method – hard:

12x – 9y = 75

-12x + 32y = 40

Add: 23y = 115

Simplify: y = 5

Substitute (original equation) : 4x – 3y = 25

4x – 3(5) = 25

4x = 40

x = 10

Solution is (10, 5)


Solve by Addition/Elimination the following systems of linear equations (see worksheet #3):

#1



#2



#3



#4



#5

systems of linear equations block 44. system of linear equations a system of equations is a set or...

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