6x + 5y + 11z = 192x + 3y + 6z = 63x − 2y + 7z = 20

6x + 5y + 11z = 192x + 3y + 6z = 63x − 2y + 7z = 20

x = 196 − 5

6 y − 116 z

6x + 5y + 11z = 192x + 3y + 6z = 63x − 2y + 7z = 20

x = 196 − 5

6 y − 116 z

2( 19

6 − 56 y − 11

6 z)+ 3y + 6z = 6

6x + 5y + 11z = 192x + 3y + 6z = 63x − 2y + 7z = 20

x = 196 − 5

6 y − 116 z

2( 19

6 − 56 y − 11

6 z)+ 3y + 6z = 6

43 y + 7

3 z = − 13

6x + 5y + 11z = 192x + 3y + 6z = 63x − 2y + 7z = 20

x = 196 − 5

6 y − 116 z

2( 19

6 − 56 y − 11

6 z)+ 3y + 6z = 6

43 y + 7

3 z = − 13

y = − 74 z − 1

4

6x + 5y + 11z = 192x + 3y + 6z = 63x − 2y + 7z = 20

x = 196 − 5

6 y − 116 z

2( 19

6 − 56 y − 11

6 z)+ 3y + 6z = 6

43 y + 7

3 z = − 13

y = − 74 z − 1

4

3( 19

6 − 56 y − 11

6 z)− 2y + 7z = 20

6x + 5y + 11z = 192x + 3y + 6z = 63x − 2y + 7z = 20

x = 196 − 5

6 y − 116 z

2( 19

6 − 56 y − 11

6 z)+ 3y + 6z = 6

43 y + 7

3 z = − 13

y = − 74 z − 1

4

3( 19

6 − 56 y − 11

6 z)− 2y + 7z = 20

− 92 y + 3

2 z = 212

6x + 5y + 11z = 192x + 3y + 6z = 63x − 2y + 7z = 20

x = 196 − 5

6 y − 116 z

2( 19

6 − 56 y − 11

6 z)+ 3y + 6z = 6

43 y + 7

3 z = − 13

y = − 74 z − 1

4

3( 19

6 − 56 y − 11

6 z)− 2y + 7z = 20

− 92 y + 3

2 z = 212

− 92

(− 7

4 z − 14

)+ 3

2 z = 212

6x + 5y + 11z = 192x + 3y + 6z = 63x − 2y + 7z = 20

x = 196 − 5

6 y − 116 z

2( 19

6 − 56 y − 11

6 z)+ 3y + 6z = 6

43 y + 7

3 z = − 13

y = − 74 z − 1

4

3( 19

6 − 56 y − 11

6 z)− 2y + 7z = 20

− 92 y + 3

2 z = 212

− 92

(− 7

4 z − 14

)+ 3

2 z = 212 z = 1

6x + 5y + 11z = 192x + 3y + 6z = 63x − 2y + 7z = 20

x = 196 − 5

6 y − 116 z

2( 19

6 − 56 y − 11

6 z)+ 3y + 6z = 6

43 y + 7

3 z = − 13

y = − 74 z − 1

4

3( 19

6 − 56 y − 11

6 z)− 2y + 7z = 20

− 92 y + 3

2 z = 212

− 92

(− 7

4 z − 14

)+ 3

2 z = 212 z = 1

y = −2

6x + 5y + 11z = 192x + 3y + 6z = 63x − 2y + 7z = 20

x = 196 − 5

6 y − 116 z

2( 19

6 − 56 y − 11

6 z)+ 3y + 6z = 6

43 y + 7

3 z = − 13

y = − 74 z − 1

4

3( 19

6 − 56 y − 11

6 z)− 2y + 7z = 20

− 92 y + 3

2 z = 212

− 92

(− 7

4 z − 14

)+ 3

2 z = 212 z = 1

y = −2

x = 3

Conclusion

The substitution method of solving systems of linearequations may work well if there are only 2 variables.If there are 3 or more variables, it can become a nightmare.

Conclusion

The substitution method of solving systems of linearequations may work well if there are only 2 variables.If there are 3 or more variables, it can become a nightmare.

2x + 3y − 4z = 6−2x + y + 2z = 46x + 9y − 5z = 123x − 2y − 3z = 1

2x + 3y − 4z = 6−2x + y + 2z = 46x + 9y − 5z = 123x − 2y − 3z = 1

2x + 3y − 4z = 6−2x + y + 2z = 4

4y − 2z = 10

2x + 3y − 4z = 6−2x + y + 2z = 46x + 9y − 5z = 123x − 2y − 3z = 1

2x + 3y − 4z = 6−2x + y + 2z = 4

4y − 2z = 10

6x + 9y − 5z = 12−6x + 4y + 6z = −2

13y + z = 10

2x + 3y − 4z = 6−2x + y + 2z = 46x + 9y − 5z = 123x − 2y − 3z = 1

2x + 3y − 4z = 6−2x + y + 2z = 4

4y − 2z = 10

6x + 9y − 5z = 12−6x + 4y + 6z = −2

13y + z = 10

4y − 2z = 1026y + 2z = 2030y = 30

2x + 3y − 4z = 6−2x + y + 2z = 46x + 9y − 5z = 123x − 2y − 3z = 1

2x + 3y − 4z = 6−2x + y + 2z = 4

4y − 2z = 10

6x + 9y − 5z = 12−6x + 4y + 6z = −2

13y + z = 10

4y − 2z = 1026y + 2z = 2030y = 30

y = 1

2x + 3y − 4z = 6−2x + y + 2z = 46x + 9y − 5z = 123x − 2y − 3z = 1

2x + 3y − 4z = 6−2x + y + 2z = 4

4y − 2z = 10

6x + 9y − 5z = 12−6x + 4y + 6z = −2

13y + z = 10

4y − 2z = 1026y + 2z = 2030y = 30

y = 1z = 10 − 13y

2x + 3y − 4z = 6−2x + y + 2z = 46x + 9y − 5z = 123x − 2y − 3z = 1

2x + 3y − 4z = 6−2x + y + 2z = 4

4y − 2z = 10

6x + 9y − 5z = 12−6x + 4y + 6z = −2

13y + z = 10

4y − 2z = 1026y + 2z = 2030y = 30

y = 1z = 10 − 13y

z = −3

2x + 3y − 4z = 6−2x + y + 2z = 46x + 9y − 5z = 123x − 2y − 3z = 1

2x + 3y − 4z = 6−2x + y + 2z = 4

4y − 2z = 10

6x + 9y − 5z = 12−6x + 4y + 6z = −2

13y + z = 10

4y − 2z = 1026y + 2z = 2030y = 30

y = 1z = 10 − 13y

z = −3

x = 16 (12 − 9y + 5z)

2x + 3y − 4z = 6−2x + y + 2z = 46x + 9y − 5z = 123x − 2y − 3z = 1

2x + 3y − 4z = 6−2x + y + 2z = 4

4y − 2z = 10

6x + 9y − 5z = 12−6x + 4y + 6z = −2

13y + z = 10

4y − 2z = 1026y + 2z = 2030y = 30

y = 1z = 10 − 13y

z = −3

x = 16 (12 − 9y + 5z) x = −2

2x + 3y − 4z = 6−2x + y + 2z = 46x + 9y − 5z = 123x − 2y − 3z = 1

2x + 3y − 4z = 6−2x + y + 2z = 4

4y − 2z = 10

6x + 9y − 5z = 12−6x + 4y + 6z = −2

13y + z = 10

4y − 2z = 1026y + 2z = 2030y = 30

y = 1z = 10 − 13y

z = −3

x = 16 (12 − 9y + 5z) x = −2

2x + 3y − 4z ?= 6

2x + 3y − 4z = 6−2x + y + 2z = 46x + 9y − 5z = 123x − 2y − 3z = 1

2x + 3y − 4z = 6−2x + y + 2z = 4

4y − 2z = 10

6x + 9y − 5z = 12−6x + 4y + 6z = −2

13y + z = 10

4y − 2z = 1026y + 2z = 2030y = 30

y = 1z = 10 − 13y

z = −3

x = 16 (12 − 9y + 5z) x = −2

2x + 3y − 4z ?= 6

2(−2) + 3(1)− 4(−3) = 11

2x + 3y − 4z = 6−2x + y + 2z = 46x + 9y − 5z = 123x − 2y − 3z = 1

2x + 3y − 4z = 6−2x + y + 2z = 4

4y − 2z = 10

6x + 9y − 5z = 12−6x + 4y + 6z = −2

13y + z = 10

4y − 2z = 1026y + 2z = 2030y = 30

y = 1z = 10 − 13y

z = −3

x = 16 (12 − 9y + 5z) x = −2

2x + 3y − 4z ?= 6

2(−2) + 3(1)− 4(−3) = 11

???

Conclusion

The addition method of solving systems of linearequations sometimes works well if there are only 2 or 3equations.However, it’s easy to get lost if you don’t keep organized.Far worse, if you just work haphazardly, you can get wronganswers!

Conclusion

The addition method of solving systems of linearequations sometimes works well if there are only 2 or 3equations.However, it’s easy to get lost if you don’t keep organized.Far worse, if you just work haphazardly, you can get wronganswers!

Conclusion

The addition method of solving systems of linearequations sometimes works well if there are only 2 or 3equations.However, it’s easy to get lost if you don’t keep organized.Far worse, if you just work haphazardly, you can get wronganswers!

What We Want

Our DesireWe want a system for solving systems of linear equations that

produces all solutions to the system.never produces fake solutions.is systematic and organized.never requires guesswork to know the next move.

What We Want

Our DesireWe want a system for solving systems of linear equations that

produces all solutions to the system.never produces fake solutions.is systematic and organized.never requires guesswork to know the next move.

What We Want

Our DesireWe want a system for solving systems of linear equations that

produces all solutions to the system.never produces fake solutions.is systematic and organized.never requires guesswork to know the next move.

What We Want

Our DesireWe want a system for solving systems of linear equations that

produces all solutions to the system.never produces fake solutions.is systematic and organized.never requires guesswork to know the next move.

What We Want

Our DesireWe want a system for solving systems of linear equations that

produces all solutions to the system.never produces fake solutions.is systematic and organized.never requires guesswork to know the next move.