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3.6-Linear Systems with Three Variables

Ex:

A: 3x + 2y + 2=6

B: -2x-y + 3z = 0

C: -5x + 2y + 2z = -1

Substitution Method

Steps for Solving:

1) Choose one variable from any of the three equations and solve for it.

2) Plug the variable found in step one into one of the other two equations (note: cannot be an equation that has already been used in step one).

3) Plug the variable found in step one into the remaining equation.

4) Solve the two-variable system that you have just created.

5) Finally, solve for the third and final variable.

A= 2=6-3x-2y

A into C

-5x+2y+2(6-3x-2y)

-5x+2y+12-6x-4y=-1

-11x-2y=-13

A into B

-2x-y+3(6-3x-2y)

-2x-y+18-9x-6y

-11x-7y=-18

Solve the 2-variable system

-11x-2y=-13

-11x-7y=-18

-5y=-5

y=1

-11x-2(1)=13

+2 +2

-11x=-11

x=1

3(1)+2(1)+2=6

5+2=6

z=1

Linear Combination Method

A: 3x+3y-2=8

B:-3x+4y+5z=-14

C:x-3y+4z=-14

Steps for Solving:

1) Choose a variable to remove from the equation-for example the variable x.

2) In order to remove the variable using the linear combination method, use equations A and B.

3) After you have removed the variable successfully, use two different combinations of equations to remove another variable-for example-equations A and C or equations B and C.

A 3x+2y-z=8

B -3x+4y+5z=-14

6y+4z=-6

B -3x+4y+5z=-14

C 3x-9y+12z=-42

-5y+17z=-56

5(6y+4z=-6)

6(-5y+17z=56

30y+20z=-30

-30y+102z=-336

122z=-366

z=-366

122

z=-3

6y+4(-3)=-6

6y-12=-6

6y=6

y=6

y=1

3x+2(1)+3=8

3x+2+3=8

3x+5=8

3x=8-5

3x=3

x=3

x=1